WEBVTT 00:00:00.000 --> 00:00:02.435 align:middle line:90% [SQUEAKING] 00:00:02.435 --> 00:00:03.896 align:middle line:90% [RUSTLING] 00:00:03.896 --> 00:00:05.357 align:middle line:90% [CLICKING] 00:00:05.357 --> 00:00:12.180 align:middle line:90% 00:00:12.180 --> 00:00:13.270 align:middle line:90% RAFAEL JARAMILLO: Hello. 00:00:13.270 --> 00:00:14.220 align:middle line:90% Happy Monday. 00:00:14.220 --> 00:00:16.770 align:middle line:90% And welcome to lecture 28. 00:00:16.770 --> 00:00:21.480 align:middle line:84% We're going to continue working on statistical thermodynamics, 00:00:21.480 --> 00:00:24.240 align:middle line:84% and we'll start with the Boltzmann hypothesis. 00:00:24.240 --> 00:00:31.880 align:middle line:90% 00:00:31.880 --> 00:00:33.090 align:middle line:90% All right. 00:00:33.090 --> 00:00:41.460 align:middle line:84% And what Gibbs is to classical thermal, 00:00:41.460 --> 00:00:47.910 align:middle line:90% Boltzmann is to statistically. 00:00:47.910 --> 00:00:52.960 align:middle line:90% 00:00:52.960 --> 00:00:55.780 align:middle line:84% And so these are really giants, and the reason why their names 00:00:55.780 --> 00:00:57.190 align:middle line:90% are all over everything. 00:00:57.190 --> 00:01:00.115 align:middle line:84% My understanding is that Gibbs was a pretty straight shooter, 00:01:00.115 --> 00:01:04.209 align:middle line:84% and his biography is a little bit on the boring side. 00:01:04.209 --> 00:01:05.620 align:middle line:90% That's my very quick read of it. 00:01:05.620 --> 00:01:08.870 align:middle line:84% Boltzmann was a much more turbulent individual. 00:01:08.870 --> 00:01:12.310 align:middle line:84% And so it may not have been fun to be Boltzmann, 00:01:12.310 --> 00:01:14.240 align:middle line:84% but his biography is a lot more interesting. 00:01:14.240 --> 00:01:17.170 align:middle line:84% So if you have a moment, you might read a little bit 00:01:17.170 --> 00:01:19.480 align:middle line:84% about his life, anyway, but we're just 00:01:19.480 --> 00:01:21.642 align:middle line:90% going to focus on the science. 00:01:21.642 --> 00:01:22.600 align:middle line:90% So here's the preamble. 00:01:22.600 --> 00:01:28.920 align:middle line:90% 00:01:28.920 --> 00:01:32.715 align:middle line:84% We have this quantity, which we counted microstates, omega. 00:01:32.715 --> 00:01:37.910 align:middle line:90% 00:01:37.910 --> 00:01:40.750 align:middle line:90% It describes the stability. 00:01:40.750 --> 00:01:43.530 align:middle line:84% We haven't put it in these terms, 00:01:43.530 --> 00:01:47.880 align:middle line:84% but it follows from everything we've been doing. 00:01:47.880 --> 00:01:50.160 align:middle line:84% Omega describes the stability of a macrostate 00:01:50.160 --> 00:02:03.940 align:middle line:84% because the state with maximum number of microstates 00:02:03.940 --> 00:02:04.915 align:middle line:90% will appear. 00:02:04.915 --> 00:02:17.950 align:middle line:90% 00:02:17.950 --> 00:02:18.450 align:middle line:90% All right. 00:02:18.450 --> 00:02:21.960 align:middle line:84% So this is an observation from what we've done so far. 00:02:21.960 --> 00:02:25.050 align:middle line:84% And I hope this is sensible to you. 00:02:25.050 --> 00:02:27.300 align:middle line:84% That the state with the maximum number of microstates 00:02:27.300 --> 00:02:28.830 align:middle line:90% will appear the most stable. 00:02:28.830 --> 00:02:32.070 align:middle line:84% It's the most likely, and if you find yourself in that state, 00:02:32.070 --> 00:02:34.150 align:middle line:84% you're very unlikely to get out of that state. 00:02:34.150 --> 00:02:37.680 align:middle line:84% So it's going to appear to have this property of stability. 00:02:37.680 --> 00:02:40.650 align:middle line:84% And if you remember, towards the beginning of this class, 00:02:40.650 --> 00:02:46.830 align:middle line:84% we defined equilibrium as having this property of stability. 00:02:46.830 --> 00:02:49.060 align:middle line:84% And so there's some connection there. 00:02:49.060 --> 00:02:52.210 align:middle line:84% And so Boltzmann made that connection with his hypothesis. 00:02:52.210 --> 00:02:55.770 align:middle line:90% 00:02:55.770 --> 00:03:01.920 align:middle line:84% And his hypothesis is that the entropy is a function 00:03:01.920 --> 00:03:03.390 align:middle line:90% of the number of microstates. 00:03:03.390 --> 00:03:05.610 align:middle line:84% So the entropy of a macrostate is a function 00:03:05.610 --> 00:03:07.710 align:middle line:84% of the number of microstates corresponding 00:03:07.710 --> 00:03:08.850 align:middle line:90% to that macrostate. 00:03:08.850 --> 00:03:11.970 align:middle line:84% And more than that, he hypothesized 00:03:11.970 --> 00:03:31.810 align:middle line:84% that entropy is a monotonically increasing function of omega. 00:03:31.810 --> 00:03:34.100 align:middle line:84% So this is a monotonically increasing function. 00:03:34.100 --> 00:03:45.980 align:middle line:84% So what that means is that max S means max omega. 00:03:45.980 --> 00:03:47.150 align:middle line:90% So we're going to use that. 00:03:47.150 --> 00:03:49.040 align:middle line:90% Max entropy means max omega. 00:03:49.040 --> 00:03:52.520 align:middle line:90% So that's a hypothesis. 00:03:52.520 --> 00:03:53.020 align:middle line:90% All right. 00:03:53.020 --> 00:03:56.440 align:middle line:84% So we're going to consider the form of that function. 00:03:56.440 --> 00:04:01.240 align:middle line:84% This is pretty easily done, consider 00:04:01.240 --> 00:04:15.270 align:middle line:84% two isolated systems, system A and system 00:04:15.270 --> 00:04:22.860 align:middle line:90% B. System A is some stuff. 00:04:22.860 --> 00:04:25.920 align:middle line:90% I don't know what it is. 00:04:25.920 --> 00:04:31.635 align:middle line:84% And it has number of microstates omega sub a, 00:04:31.635 --> 00:04:36.295 align:middle line:84% and it has entropy extensive S of a. 00:04:36.295 --> 00:04:38.530 align:middle line:84% And system B is some other stuff. 00:04:38.530 --> 00:04:41.710 align:middle line:84% It's isolated from system A. It's over on a different room, 00:04:41.710 --> 00:04:43.420 align:middle line:84% on a different shelf, what have you, 00:04:43.420 --> 00:04:52.395 align:middle line:84% and it has microstates omega b and entropy S of b. 00:04:52.395 --> 00:04:55.930 align:middle line:90% 00:04:55.930 --> 00:05:03.780 align:middle line:84% So first of all, entropy is extensive, 00:05:03.780 --> 00:05:07.535 align:middle line:84% which means that the total entropy in the system-- 00:05:07.535 --> 00:05:10.690 align:middle line:90% 00:05:10.690 --> 00:05:12.250 align:middle line:84% like any other extensive quantity, 00:05:12.250 --> 00:05:16.168 align:middle line:84% what should the entropy of the whole be? 00:05:16.168 --> 00:05:17.085 align:middle line:90% I'll label that total. 00:05:17.085 --> 00:05:20.560 align:middle line:90% 00:05:20.560 --> 00:05:24.070 align:middle line:84% STUDENT: Be the entropy of a plus the entropy b. 00:05:24.070 --> 00:05:25.600 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:05:25.600 --> 00:05:28.120 align:middle line:84% Like any other extensive thing, like marbles in a jar 00:05:28.120 --> 00:05:34.370 align:middle line:84% or moles of material, you just add them. 00:05:34.370 --> 00:05:41.590 align:middle line:84% But the total number of microstates is-- 00:05:41.590 --> 00:05:42.995 align:middle line:90% I'll call it combinatoric. 00:05:42.995 --> 00:05:44.245 align:middle line:90% I don't know if that's a word. 00:05:44.245 --> 00:05:47.110 align:middle line:90% 00:05:47.110 --> 00:05:52.400 align:middle line:84% What I mean is, the total number of microstates 00:05:52.400 --> 00:05:56.980 align:middle line:84% available if you consider both systems is what? 00:05:56.980 --> 00:06:00.490 align:middle line:84% You can have microstate 1 and all of these, 00:06:00.490 --> 00:06:03.130 align:middle line:84% and you can have microstate 2 and all of these, 00:06:03.130 --> 00:06:05.840 align:middle line:84% and microstate 3 and all of these, and so forth. 00:06:05.840 --> 00:06:08.410 align:middle line:84% So what's the expression for the total number of microstates 00:06:08.410 --> 00:06:11.740 align:middle line:84% considering both systems at the same time? 00:06:11.740 --> 00:06:17.960 align:middle line:90% 00:06:17.960 --> 00:06:19.822 align:middle line:84% STUDENT: Is it the product of a and b? 00:06:19.822 --> 00:06:21.530 align:middle line:84% RAFAEL JARAMILLO: It's the product, yeah. 00:06:21.530 --> 00:06:24.210 align:middle line:90% 00:06:24.210 --> 00:06:24.780 align:middle line:90% All right. 00:06:24.780 --> 00:06:31.600 align:middle line:84% So entropy is additive, but omega is multiplicative. 00:06:31.600 --> 00:06:38.410 align:middle line:84% So that means f of omega total, which 00:06:38.410 --> 00:06:52.300 align:middle line:84% is f of omega a times omega b, is f omega a plus f omega b. 00:06:52.300 --> 00:06:57.810 align:middle line:84% We're using both of these properties now. 00:06:57.810 --> 00:07:01.440 align:middle line:84% So who knows a function that has this property, 00:07:01.440 --> 00:07:07.258 align:middle line:84% the function of the product is the sum of the Function 00:07:07.258 --> 00:07:07.800 align:middle line:90% STUDENT: Log? 00:07:07.800 --> 00:07:08.480 align:middle line:90% STUDENT: Logs? 00:07:08.480 --> 00:07:10.840 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:07:10.840 --> 00:07:13.670 align:middle line:84% So we come up with this conclusion, 00:07:13.670 --> 00:07:16.580 align:middle line:84% which is that this function we're looking for, 00:07:16.580 --> 00:07:22.830 align:middle line:84% which is entropy, is proportional to log of omega. 00:07:22.830 --> 00:07:26.190 align:middle line:84% And we're going to right now just have a-- 00:07:26.190 --> 00:07:27.540 align:middle line:90% we have a pre factor out there. 00:07:27.540 --> 00:07:29.070 align:middle line:90% It doesn't change. 00:07:29.070 --> 00:07:31.070 align:middle line:84% And so we're going to call that pre factor k, 00:07:31.070 --> 00:07:32.900 align:middle line:84% and we'll put a b under it because it 00:07:32.900 --> 00:07:35.730 align:middle line:90% be Boltzmann constant. 00:07:35.730 --> 00:07:45.530 align:middle line:84% We don't know what it is yet, Boltzmann's constant. 00:07:45.530 --> 00:07:47.675 align:middle line:84% And this is known as Boltzmann's entropy formula. 00:07:47.675 --> 00:07:59.140 align:middle line:90% 00:07:59.140 --> 00:08:03.210 align:middle line:84% So just like that, we have a pretty simple-looking equation 00:08:03.210 --> 00:08:06.150 align:middle line:84% that gives us the entropy for a system as a function 00:08:06.150 --> 00:08:07.500 align:middle line:90% of the number of microstates. 00:08:07.500 --> 00:08:12.778 align:middle line:90% 00:08:12.778 --> 00:08:13.570 align:middle line:90% That's pretty cool. 00:08:13.570 --> 00:08:17.860 align:middle line:90% 00:08:17.860 --> 00:08:18.360 align:middle line:90% All right. 00:08:18.360 --> 00:08:20.527 align:middle line:84% So we still don't know what Boltzmann's constant is, 00:08:20.527 --> 00:08:21.570 align:middle line:90% but that's OK. 00:08:21.570 --> 00:08:23.820 align:middle line:84% So let's see what some implications of that are. 00:08:23.820 --> 00:08:26.190 align:middle line:84% We'll start with the same configurational entropy, which 00:08:26.190 --> 00:08:28.140 align:middle line:84% we have talked about throughout the term. 00:08:28.140 --> 00:08:34.080 align:middle line:90% 00:08:34.080 --> 00:08:40.679 align:middle line:84% Configurational entropy, so what is? 00:08:40.679 --> 00:08:44.160 align:middle line:84% It's a number of ways to configure a system and space. 00:08:44.160 --> 00:08:46.690 align:middle line:90% 00:08:46.690 --> 00:08:56.920 align:middle line:90% So I'm going to draw a grid. 00:08:56.920 --> 00:08:59.320 align:middle line:84% This is going to be simply suggestive. 00:08:59.320 --> 00:09:01.570 align:middle line:84% We're not going to analyze my drawing because I'm just 00:09:01.570 --> 00:09:02.870 align:middle line:90% going to be rough about it. 00:09:02.870 --> 00:09:05.020 align:middle line:84% Let's just put a particle here and a particle here 00:09:05.020 --> 00:09:08.338 align:middle line:84% and a particle here and a particle here. 00:09:08.338 --> 00:09:09.880 align:middle line:84% What we're going to do is we're going 00:09:09.880 --> 00:09:28.275 align:middle line:84% to count the number of ways to distribute n molecules into r 00:09:28.275 --> 00:09:28.775 align:middle line:90% boxes. 00:09:28.775 --> 00:09:33.780 align:middle line:90% 00:09:33.780 --> 00:09:45.030 align:middle line:84% And we're going to have those boxes sufficiently small such 00:09:45.030 --> 00:09:56.690 align:middle line:84% that no box has more than one molecule. 00:09:56.690 --> 00:09:59.650 align:middle line:90% 00:09:59.650 --> 00:10:05.850 align:middle line:84% So we're dividing space up into tiny, tiny, tiny little voxels 00:10:05.850 --> 00:10:07.620 align:middle line:84% Well, we know how to do this already. 00:10:07.620 --> 00:10:10.080 align:middle line:84% We talked about this the other time. 00:10:10.080 --> 00:10:19.370 align:middle line:84% This is just r choose n, and that's r factorial-- 00:10:19.370 --> 00:10:22.040 align:middle line:84% wait, sorry-- over n factorial-- there's an error in the notes 00:10:22.040 --> 00:10:23.240 align:middle line:90% there-- 00:10:23.240 --> 00:10:27.320 align:middle line:90% r minus n factorial, r choose n. 00:10:27.320 --> 00:10:28.647 align:middle line:90% Good. 00:10:28.647 --> 00:10:30.230 align:middle line:84% So now what we're going to do is we're 00:10:30.230 --> 00:10:34.500 align:middle line:84% going to let the total number of boxes 00:10:34.500 --> 00:10:39.810 align:middle line:84% be the total volume divided by some little volume b. 00:10:39.810 --> 00:10:50.635 align:middle line:84% And so this is total volume and this is a little voxel. 00:10:50.635 --> 00:10:53.390 align:middle line:90% 00:10:53.390 --> 00:10:58.670 align:middle line:84% And we can say corresponds to volume-- 00:10:58.670 --> 00:11:05.930 align:middle line:90% 00:11:05.930 --> 00:11:08.990 align:middle line:84% we can say this corresponds to the volume of a molecule. 00:11:08.990 --> 00:11:11.430 align:middle line:84% It's a tiny, tiny little amount of space, 00:11:11.430 --> 00:11:15.350 align:middle line:84% but it enforces our condition here that the boxes cannot have 00:11:15.350 --> 00:11:16.490 align:middle line:90% more than one molecule. 00:11:16.490 --> 00:11:19.390 align:middle line:90% 00:11:19.390 --> 00:11:23.830 align:middle line:90% So now it can be shown-- 00:11:23.830 --> 00:11:25.780 align:middle line:90% you will apiece that-- 00:11:25.780 --> 00:11:33.850 align:middle line:84% can be shown that the log of r choose n-- 00:11:33.850 --> 00:11:36.280 align:middle line:84% you're going to take the log of omega, that's 00:11:36.280 --> 00:11:38.477 align:middle line:84% the Boltzmann hypothesis, Boltzmann entropy formula 00:11:38.477 --> 00:11:40.060 align:middle line:84% thing, so you're going to take the log 00:11:40.060 --> 00:11:44.545 align:middle line:84% of this binomial coefficient, the log of r 00:11:44.545 --> 00:11:55.190 align:middle line:84% choose n is approximately n log r for r 00:11:55.190 --> 00:11:58.790 align:middle line:90% very, very much larger than n. 00:11:58.790 --> 00:12:01.670 align:middle line:84% So r very, very much larger than an n corresponds to-- 00:12:01.670 --> 00:12:03.650 align:middle line:90% let's say a gas. 00:12:03.650 --> 00:12:05.480 align:middle line:90% Most of space is empty. 00:12:05.480 --> 00:12:10.340 align:middle line:84% There's a much larger number of voxels of space 00:12:10.340 --> 00:12:12.555 align:middle line:84% than there are molecules that fill them. 00:12:12.555 --> 00:12:13.430 align:middle line:90% So that's like a gas. 00:12:13.430 --> 00:12:16.000 align:middle line:90% 00:12:16.000 --> 00:12:17.941 align:middle line:90% And this is a-- 00:12:17.941 --> 00:12:21.030 align:middle line:90% 00:12:21.030 --> 00:12:23.795 align:middle line:84% that's a problem on a current PSET. 00:12:23.795 --> 00:12:28.510 align:middle line:90% 00:12:28.510 --> 00:12:30.040 align:middle line:90% So far so good. 00:12:30.040 --> 00:12:41.080 align:middle line:84% Now what we're going to do, now let system 00:12:41.080 --> 00:12:49.140 align:middle line:90% expand from volume to 2 volume. 00:12:49.140 --> 00:12:52.670 align:middle line:84% So we're going to do an expansion of this gas, 00:12:52.670 --> 00:12:54.590 align:middle line:84% and we're going to calculate delta 00:12:54.590 --> 00:12:59.510 align:middle line:84% S. We're going to do this using statistical thermodynamics now. 00:12:59.510 --> 00:13:03.630 align:middle line:84% We're not going to do it the way we did a month and a 1/2 ago. 00:13:03.630 --> 00:13:09.370 align:middle line:84% You have Boltzmann's constant times n, and let's see. 00:13:09.370 --> 00:13:16.135 align:middle line:84% Log 2 volume over b minus log volume over b. 00:13:16.135 --> 00:13:20.140 align:middle line:90% 00:13:20.140 --> 00:13:22.960 align:middle line:84% And we can collect terms and simplify. 00:13:22.960 --> 00:13:39.360 align:middle line:84% This is kbn log 2 v over v equals kbn log 2. 00:13:39.360 --> 00:13:47.420 align:middle line:90% 00:13:47.420 --> 00:14:03.700 align:middle line:84% So this here, kbn, this is R n equals Avogadro's number. 00:14:03.700 --> 00:14:06.420 align:middle line:90% 00:14:06.420 --> 00:14:09.660 align:middle line:84% We recall this from the classical derivation, 00:14:09.660 --> 00:14:13.890 align:middle line:84% isothermal expansion of an ideal gas. 00:14:13.890 --> 00:14:30.070 align:middle line:84% This is the same result for isothermal expansion 00:14:30.070 --> 00:14:32.470 align:middle line:90% of ideal gas. 00:14:32.470 --> 00:14:41.620 align:middle line:84% Delta S equals R log v final over v initial. 00:14:41.620 --> 00:14:45.650 align:middle line:90% 00:14:45.650 --> 00:14:47.900 align:middle line:84% So this is one way to start identifying what 00:14:47.900 --> 00:14:50.480 align:middle line:90% that Boltzmann constant is. 00:14:50.480 --> 00:14:56.780 align:middle line:84% Boltzmann's constant is R divided by Avogadro's number. 00:14:56.780 --> 00:14:57.305 align:middle line:90% That's cool. 00:14:57.305 --> 00:15:01.220 align:middle line:90% 00:15:01.220 --> 00:15:02.060 align:middle line:90% That's neat. 00:15:02.060 --> 00:15:05.030 align:middle line:84% We've done this weird statistical thing, 00:15:05.030 --> 00:15:10.130 align:middle line:84% and good, old Ludwig came up with this. 00:15:10.130 --> 00:15:15.140 align:middle line:84% And we find that when we calculate a simple case, 00:15:15.140 --> 00:15:17.420 align:middle line:84% we get functionally the same thing as when 00:15:17.420 --> 00:15:19.940 align:middle line:84% we calculated this case, when we didn't know anything 00:15:19.940 --> 00:15:20.990 align:middle line:90% about molecules. 00:15:20.990 --> 00:15:24.600 align:middle line:84% And we were just dealing with classical thermodynamics. 00:15:24.600 --> 00:15:27.220 align:middle line:84% We can make the connection via the coefficients. 00:15:27.220 --> 00:15:27.810 align:middle line:90% That's cool. 00:15:27.810 --> 00:15:32.590 align:middle line:90% 00:15:32.590 --> 00:15:33.090 align:middle line:90% Great. 00:15:33.090 --> 00:15:33.990 align:middle line:90% Questions about this? 00:15:33.990 --> 00:15:36.165 align:middle line:84% Because I'm going to move on to the next thing 00:15:36.165 --> 00:15:37.230 align:middle line:90% that Boltzmann did. 00:15:37.230 --> 00:15:39.685 align:middle line:90% 00:15:39.685 --> 00:15:41.060 align:middle line:84% The next thing we're going to do, 00:15:41.060 --> 00:15:44.330 align:middle line:84% and what we're really building up to here, 00:15:44.330 --> 00:15:59.400 align:middle line:84% is the maximum entropy condition, maximum entropy 00:15:59.400 --> 00:16:08.600 align:middle line:84% condition and the Boltzmann distribution. 00:16:08.600 --> 00:16:13.640 align:middle line:90% 00:16:13.640 --> 00:16:17.162 align:middle line:84% So we get Boltzmann hypothesis, Boltzmann entropy formula. 00:16:17.162 --> 00:16:19.120 align:middle line:84% And this is going to be Boltzmann distribution, 00:16:19.120 --> 00:16:25.030 align:middle line:84% this Boltzmann's constant Boltzmann's name is everywhere. 00:16:25.030 --> 00:16:34.970 align:middle line:84% We're going to set this up and do most of it, 00:16:34.970 --> 00:16:37.310 align:middle line:84% but this will carry over into lecture 29. 00:16:37.310 --> 00:16:39.740 align:middle line:84% It's that essential that it's worth taking the time. 00:16:39.740 --> 00:16:43.680 align:middle line:90% 00:16:43.680 --> 00:16:44.180 align:middle line:90% All right. 00:16:44.180 --> 00:16:45.263 align:middle line:90% So what we're going to do? 00:16:45.263 --> 00:17:02.880 align:middle line:84% We're going to consider n total particles distributed 00:17:02.880 --> 00:17:14.764 align:middle line:84% over r states according to the occupation numbers. 00:17:14.764 --> 00:17:21.540 align:middle line:90% 00:17:21.540 --> 00:17:24.490 align:middle line:84% We have these occupation numbers from last time. 00:17:24.490 --> 00:17:24.990 align:middle line:90% All right. 00:17:24.990 --> 00:17:31.258 align:middle line:84% That's a set of numbers, n of 1, n of 2, and all over-- 00:17:31.258 --> 00:17:32.300 align:middle line:90% all the way up to n of r. 00:17:32.300 --> 00:17:35.690 align:middle line:84% So how many states are-- how many particles in each state? 00:17:35.690 --> 00:17:40.980 align:middle line:84% And we're going to calculate the entropy of this thing. 00:17:40.980 --> 00:17:45.570 align:middle line:84% So again, taking from last time, from our last lecture, 00:17:45.570 --> 00:17:53.440 align:middle line:84% is going to be kb time to log n of total 00:17:53.440 --> 00:18:00.800 align:middle line:84% over product n of i, everything factorial. 00:18:00.800 --> 00:18:04.260 align:middle line:90% 00:18:04.260 --> 00:18:15.290 align:middle line:84% Now, I'm going to use Sterling's approximation 00:18:15.290 --> 00:18:20.960 align:middle line:84% to get to the next line, which is going to be kb, 00:18:20.960 --> 00:18:35.400 align:middle line:84% n total log n total minus n total minus sum n of i log 00:18:35.400 --> 00:18:41.150 align:middle line:90% n of i plus sum n of i. 00:18:41.150 --> 00:18:46.780 align:middle line:90% 00:18:46.780 --> 00:18:48.920 align:middle line:84% So I use this Sterling approximation, 00:18:48.920 --> 00:18:55.446 align:middle line:84% and then I'm going to use the fact, the sum over i. 00:18:55.446 --> 00:18:58.420 align:middle line:84% n of ni equals n total, so the sum 00:18:58.420 --> 00:19:02.260 align:middle line:84% of where all the particles are equals all the particles. 00:19:02.260 --> 00:19:04.200 align:middle line:90% So I can simplify a little bit. 00:19:04.200 --> 00:19:13.790 align:middle line:84% This equals kb n total log n total 00:19:13.790 --> 00:19:22.600 align:middle line:84% minus sum over i, n of i log n of i. 00:19:22.600 --> 00:19:26.221 align:middle line:84% And now I can condense this a little bit. 00:19:26.221 --> 00:19:30.460 align:middle line:84% I'm going to split the numerator and denominator, 00:19:30.460 --> 00:19:39.414 align:middle line:84% figure the minus sign, and I get a log n of i over n total. 00:19:39.414 --> 00:19:41.460 align:middle line:84% So so far, just playing with numbers. 00:19:41.460 --> 00:19:44.600 align:middle line:84% So I have the entropy is a sum for n 00:19:44.600 --> 00:19:47.340 align:middle line:90% of i log n of i over n total. 00:19:47.340 --> 00:19:53.740 align:middle line:90% 00:19:53.740 --> 00:19:56.710 align:middle line:84% By the way, this kind of looks like x log x, doesn't it? 00:19:56.710 --> 00:20:01.730 align:middle line:84% It kind of looks like our ideal entropy formula. 00:20:01.730 --> 00:20:03.710 align:middle line:84% Anyway, just a passing observation. 00:20:03.710 --> 00:20:06.650 align:middle line:90% 00:20:06.650 --> 00:20:07.150 align:middle line:90% All right. 00:20:07.150 --> 00:20:08.980 align:middle line:90% So that's fine. 00:20:08.980 --> 00:20:11.050 align:middle line:90% Here's the science inside. 00:20:11.050 --> 00:20:29.510 align:middle line:84% The distribution of occupation numbers, the distribution 00:20:29.510 --> 00:20:40.540 align:middle line:84% of occupation numbers is an unconstrained internal 00:20:40.540 --> 00:20:41.040 align:middle line:90% variable. 00:20:41.040 --> 00:20:49.070 align:middle line:90% 00:20:49.070 --> 00:20:52.250 align:middle line:84% That means that those particles are going to fluctuate. 00:20:52.250 --> 00:20:56.070 align:middle line:84% They're going to fluctuate in and out of different states, 00:20:56.070 --> 00:20:58.630 align:middle line:84% and that's an unconstrained process. 00:20:58.630 --> 00:21:02.020 align:middle line:84% So in our previous example, you had particles in a box. 00:21:02.020 --> 00:21:05.130 align:middle line:84% And I you could imagine these particles in general move. 00:21:05.130 --> 00:21:06.660 align:middle line:84% They could jump in between boxes. 00:21:06.660 --> 00:21:09.720 align:middle line:84% That's an example of jumping in between states, 00:21:09.720 --> 00:21:13.620 align:middle line:84% we're going to make this a little more general and not 00:21:13.620 --> 00:21:16.350 align:middle line:84% limited to states being positions in space. 00:21:16.350 --> 00:21:18.390 align:middle line:84% We're going to have a more general expression. 00:21:18.390 --> 00:21:23.170 align:middle line:84% We're going to say, let's say, state i minus 1. 00:21:23.170 --> 00:21:32.650 align:middle line:84% We have state i and state i plus 1, so forth. 00:21:32.650 --> 00:21:42.760 align:middle line:90% 00:21:42.760 --> 00:21:45.247 align:middle line:90% And we're going to allow that-- 00:21:45.247 --> 00:21:46.830 align:middle line:84% At any given moment, let's say there's 00:21:46.830 --> 00:21:48.760 align:middle line:90% four particles in this state. 00:21:48.760 --> 00:21:50.880 align:middle line:84% Let's say there's two particles in this state. 00:21:50.880 --> 00:21:56.370 align:middle line:84% And there was a third, but that particle jumped. 00:21:56.370 --> 00:21:59.430 align:middle line:84% It fluctuated and went over here. 00:21:59.430 --> 00:22:03.520 align:middle line:84% This is simply visually acknowledging 00:22:03.520 --> 00:22:05.110 align:middle line:90% that the states are fluctuating. 00:22:05.110 --> 00:22:08.690 align:middle line:84% The particles are fluctuating between them. 00:22:08.690 --> 00:22:19.145 align:middle line:84% These fluctuations can and will happen. 00:22:19.145 --> 00:22:25.760 align:middle line:90% 00:22:25.760 --> 00:22:39.810 align:middle line:84% So if that's happening, the maximum entropy condition, 00:22:39.810 --> 00:22:55.060 align:middle line:84% S equals S max, requires that S is what? 00:22:55.060 --> 00:22:56.850 align:middle line:90% Stationary. 00:22:56.850 --> 00:23:00.530 align:middle line:84% As we have done now so many times in this class, 00:23:00.530 --> 00:23:03.890 align:middle line:84% it has to be stationary with respect 00:23:03.890 --> 00:23:07.520 align:middle line:84% to all unconstrained internal processes. 00:23:07.520 --> 00:23:20.010 align:middle line:90% 00:23:20.010 --> 00:23:23.130 align:middle line:84% So this conceptually, where we're going is the following. 00:23:23.130 --> 00:23:24.800 align:middle line:84% We did something like this before. 00:23:24.800 --> 00:23:27.330 align:middle line:90% We've done it multiple times. 00:23:27.330 --> 00:23:32.620 align:middle line:84% When we had two systems that can exchange volume, 00:23:32.620 --> 00:23:35.080 align:middle line:84% we require the entropy to stationary with respect 00:23:35.080 --> 00:23:36.290 align:middle line:90% to that. 00:23:36.290 --> 00:23:39.550 align:middle line:84% And we got the mechanical equilibrium condition. 00:23:39.550 --> 00:23:41.200 align:middle line:90% Pressures are equal. 00:23:41.200 --> 00:23:44.440 align:middle line:84% And we had two systems that can exchange energy. 00:23:44.440 --> 00:23:50.980 align:middle line:84% We wrote out the max entropy condition, 00:23:50.980 --> 00:23:53.560 align:middle line:84% and we require that the entropy is stationary with respect 00:23:53.560 --> 00:23:55.540 align:middle line:90% to the energy exchange. 00:23:55.540 --> 00:23:57.520 align:middle line:84% And we got the thermal equilibrium condition. 00:23:57.520 --> 00:23:59.230 align:middle line:84% We got the temperatures are equal. 00:23:59.230 --> 00:24:01.990 align:middle line:84% And likewise with systems that could exchange particle number, 00:24:01.990 --> 00:24:04.582 align:middle line:84% we got chemical potential being equal. 00:24:04.582 --> 00:24:06.040 align:middle line:84% Adding that whole thing up, we call 00:24:06.040 --> 00:24:08.660 align:middle line:90% that thermodynamic equilibrium. 00:24:08.660 --> 00:24:11.220 align:middle line:84% So now we're doing something slightly different. 00:24:11.220 --> 00:24:14.000 align:middle line:84% We're requiring this entropy stationary with respect 00:24:14.000 --> 00:24:17.180 align:middle line:84% to exchange between different states. 00:24:17.180 --> 00:24:19.020 align:middle line:84% And it's pretty general right now. 00:24:19.020 --> 00:24:23.070 align:middle line:84% So a little bit vague, but there are 00:24:23.070 --> 00:24:27.030 align:middle line:84% clear similarities to what we've done earlier in the class. 00:24:27.030 --> 00:24:29.590 align:middle line:84% At least there are mathematical similarities. 00:24:29.590 --> 00:24:33.090 align:middle line:84% So ds prime-- that's write that out-- 00:24:33.090 --> 00:24:36.760 align:middle line:84% equals minus Boltzmann's constant. 00:24:36.760 --> 00:24:39.820 align:middle line:84% And what I'm doing is I'm just taking the total derivative, 00:24:39.820 --> 00:24:47.380 align:middle line:90% so L log n of i, dn of i-- 00:24:47.380 --> 00:25:01.640 align:middle line:84% see-- plus n of i over n of i d n of i minus log of n total d n 00:25:01.640 --> 00:25:07.180 align:middle line:90% of i minus n of i over n total-- 00:25:07.180 --> 00:25:12.180 align:middle line:84% just taking the total derivative of the expression 00:25:12.180 --> 00:25:13.830 align:middle line:90% on the previous slide. 00:25:13.830 --> 00:25:17.260 align:middle line:84% And this simplifies pretty readily. 00:25:17.260 --> 00:25:22.440 align:middle line:84% And I get minus Boltzmann constant 00:25:22.440 --> 00:25:32.745 align:middle line:84% and the sum over log n of i over n total dn of i. 00:25:32.745 --> 00:25:41.540 align:middle line:90% 00:25:41.540 --> 00:25:42.040 align:middle line:90% All right. 00:25:42.040 --> 00:25:44.185 align:middle line:84% Just, again, making this explicit. 00:25:44.185 --> 00:25:46.720 align:middle line:90% 00:25:46.720 --> 00:25:49.738 align:middle line:84% I just took the total derivative of this using the chain rule. 00:25:49.738 --> 00:25:54.230 align:middle line:90% 00:25:54.230 --> 00:25:54.730 align:middle line:90% All right. 00:25:54.730 --> 00:25:57.460 align:middle line:84% So that's the S, and I have all my little unconstrained 00:25:57.460 --> 00:26:01.220 align:middle line:84% internal processes here, little fluctuations 00:26:01.220 --> 00:26:04.740 align:middle line:90% between the occupation numbers. 00:26:04.740 --> 00:26:05.240 align:middle line:90% All right. 00:26:05.240 --> 00:26:07.130 align:middle line:90% Now let's apply my constraints. 00:26:07.130 --> 00:26:10.130 align:middle line:84% That's what we did before, and that's what we'll do again. 00:26:10.130 --> 00:26:12.950 align:middle line:84% And in this case, I'm going to apply isolation constraints. 00:26:12.950 --> 00:26:21.060 align:middle line:90% 00:26:21.060 --> 00:26:24.900 align:middle line:84% Isolation constraints, so I want my system to be isolated. 00:26:24.900 --> 00:26:27.330 align:middle line:84% Max entropy is the equilibrium condition 00:26:27.330 --> 00:26:28.920 align:middle line:90% for an isolated system. 00:26:28.920 --> 00:26:31.080 align:middle line:90% We remember that. 00:26:31.080 --> 00:26:34.495 align:middle line:90% So here's my little fluffy-- 00:26:34.495 --> 00:26:37.270 align:middle line:90% 00:26:37.270 --> 00:26:40.270 align:middle line:84% pink insulation here surrounding my system, 00:26:40.270 --> 00:26:50.770 align:middle line:84% and I've got system surroundings and what? 00:26:50.770 --> 00:26:54.610 align:middle line:90% The boundary is rigid. 00:26:54.610 --> 00:27:04.750 align:middle line:84% It is impermeable, and it is insulating. 00:27:04.750 --> 00:27:11.430 align:middle line:90% 00:27:11.430 --> 00:27:11.930 align:middle line:90% All right. 00:27:11.930 --> 00:27:18.520 align:middle line:84% Now we're going to allow that the states 00:27:18.520 --> 00:27:20.320 align:middle line:90% I have different energies. 00:27:20.320 --> 00:27:36.997 align:middle line:84% That e sub i be the energy per particle state I. 00:27:36.997 --> 00:27:38.830 align:middle line:84% And I don't want to just slip this in there. 00:27:38.830 --> 00:27:39.675 align:middle line:90% This is kind of new. 00:27:39.675 --> 00:27:42.830 align:middle line:90% 00:27:42.830 --> 00:27:47.390 align:middle line:84% This is kind of new for us because previously in the baby 00:27:47.390 --> 00:27:52.400 align:middle line:84% book, and even just 15 minutes ago in this example 00:27:52.400 --> 00:27:55.370 align:middle line:84% of configuration entropy, we had this idea 00:27:55.370 --> 00:27:57.470 align:middle line:84% that space was somehow flat and uniform. 00:27:57.470 --> 00:27:59.420 align:middle line:84% And the energy of each particle would not 00:27:59.420 --> 00:28:02.630 align:middle line:90% be dependent on its position. 00:28:02.630 --> 00:28:04.760 align:middle line:90% All right. 00:28:04.760 --> 00:28:06.620 align:middle line:84% So if the states are positions, maybe you 00:28:06.620 --> 00:28:08.453 align:middle line:84% have a gravitational potential, or maybe you 00:28:08.453 --> 00:28:10.280 align:middle line:90% have an electric field. 00:28:10.280 --> 00:28:13.400 align:middle line:84% Maybe this a battery, and there's an electric potential. 00:28:13.400 --> 00:28:14.990 align:middle line:84% And maybe the particles are charged, 00:28:14.990 --> 00:28:19.460 align:middle line:84% and then you can imagine energy and space becoming conflated. 00:28:19.460 --> 00:28:21.380 align:middle line:84% But more generally, there's no reason 00:28:21.380 --> 00:28:26.840 align:middle line:84% why these states have to be positions in space. 00:28:26.840 --> 00:28:30.750 align:middle line:84% They could be spin states or vibrational states. 00:28:30.750 --> 00:28:32.490 align:middle line:90% They could be rotational states. 00:28:32.490 --> 00:28:36.150 align:middle line:84% They can be anything that is distinct. 00:28:36.150 --> 00:28:38.837 align:middle line:84% Different states have a particle in general. 00:28:38.837 --> 00:28:40.920 align:middle line:84% These different states can have different energies 00:28:40.920 --> 00:28:41.940 align:middle line:90% per particle. 00:28:41.940 --> 00:28:45.960 align:middle line:84% So e sub i equals the energy per particle and state I. 00:28:45.960 --> 00:28:49.980 align:middle line:84% And so we're going to then say the total internal energy 00:28:49.980 --> 00:28:50.710 align:middle line:90% is pretty simple. 00:28:50.710 --> 00:28:53.770 align:middle line:84% It's just adding up all the energies, 00:28:53.770 --> 00:28:56.770 align:middle line:90% e sub i times n sub i. 00:28:56.770 --> 00:29:09.200 align:middle line:84% And that means that du equals e sub i dn of i. 00:29:09.200 --> 00:29:10.410 align:middle line:90% Its conservation of energy. 00:29:10.410 --> 00:29:23.480 align:middle line:90% 00:29:23.480 --> 00:29:28.501 align:middle line:84% So you're going to study a system of elastic collisions 00:29:28.501 --> 00:29:30.130 align:middle line:90% in the lab. 00:29:30.130 --> 00:29:32.050 align:middle line:84% And the individual particle energy 00:29:32.050 --> 00:29:33.460 align:middle line:84% isn't going to be changing, but I 00:29:33.460 --> 00:29:35.590 align:middle line:84% think you can trust that the total system 00:29:35.590 --> 00:29:39.760 align:middle line:84% energy for elastic collisions or rigid billiard balls, that 00:29:39.760 --> 00:29:40.520 align:middle line:90% doesn't change. 00:29:40.520 --> 00:29:43.240 align:middle line:84% So one gains and other loses and so forth. 00:29:43.240 --> 00:29:45.680 align:middle line:84% And likewise, we have the total number of particles. 00:29:45.680 --> 00:29:48.750 align:middle line:84% And this is a pretty simple expression. 00:29:48.750 --> 00:29:52.610 align:middle line:90% This is the sum n of i. 00:29:52.610 --> 00:30:03.230 align:middle line:84% And that means dn total equals the sum dn of i. 00:30:03.230 --> 00:30:04.063 align:middle line:90% I'm sorry I forgot. 00:30:04.063 --> 00:30:05.480 align:middle line:84% This is going to be 0, and this is 00:30:05.480 --> 00:30:14.935 align:middle line:84% going to be 0, conservational of mass, conservation of mass. 00:30:14.935 --> 00:30:19.170 align:middle line:90% 00:30:19.170 --> 00:30:22.780 align:middle line:84% So now I have some mathematical ways to apply these conditions. 00:30:22.780 --> 00:30:28.040 align:middle line:84% And I'll just make a note in passing, 00:30:28.040 --> 00:30:30.560 align:middle line:84% this really is beyond the scope of this class, 00:30:30.560 --> 00:30:39.985 align:middle line:84% but those who are interested, conservation of volume, 00:30:39.985 --> 00:30:41.360 align:middle line:84% you might say, what about volume? 00:30:41.360 --> 00:30:44.960 align:middle line:84% Before, two months ago, you were considering volume, energy 00:30:44.960 --> 00:30:46.100 align:middle line:90% particle number and volume. 00:30:46.100 --> 00:30:47.767 align:middle line:84% We're not touching conservational volume 00:30:47.767 --> 00:31:00.755 align:middle line:84% right now, but it's connected to the E Is being constant. 00:31:00.755 --> 00:31:06.010 align:middle line:90% 00:31:06.010 --> 00:31:09.800 align:middle line:84% When I took du, I didn't apply the chain rule 00:31:09.800 --> 00:31:11.510 align:middle line:90% and allow the E of Is to change. 00:31:11.510 --> 00:31:15.060 align:middle line:90% I assumed they were constants. 00:31:15.060 --> 00:31:17.260 align:middle line:84% And that comes from the conservational volume. 00:31:17.260 --> 00:31:21.430 align:middle line:84% So when you take quantum mechanics next semester, 00:31:21.430 --> 00:31:23.470 align:middle line:90% this comes out right away, so. 00:31:23.470 --> 00:31:23.970 align:middle line:90% All right. 00:31:23.970 --> 00:31:25.512 align:middle line:84% But we're not going to touch it here. 00:31:25.512 --> 00:31:28.690 align:middle line:90% 00:31:28.690 --> 00:31:40.170 align:middle line:84% So this is a case of constrained optimization, constrained 00:31:40.170 --> 00:31:41.220 align:middle line:90% optimization. 00:31:41.220 --> 00:31:45.010 align:middle line:90% 00:31:45.010 --> 00:31:48.580 align:middle line:84% Constraint optimization is the most important application 00:31:48.580 --> 00:31:51.430 align:middle line:84% of calculus in engineering, in business. 00:31:51.430 --> 00:31:52.930 align:middle line:84% It's in general what you do in Sloan 00:31:52.930 --> 00:31:55.870 align:middle line:84% and what you do in course 16 and course 2. 00:31:55.870 --> 00:31:58.180 align:middle line:84% A little bit less in course 3, but you 00:31:58.180 --> 00:32:00.230 align:middle line:84% should remember this from calculus. 00:32:00.230 --> 00:32:09.930 align:middle line:84% So we want to optimize that function, 00:32:09.930 --> 00:32:25.250 align:middle line:84% subject to constraints, that the energy and the total particle 00:32:25.250 --> 00:32:25.985 align:middle line:90% number are fixed. 00:32:25.985 --> 00:32:28.933 align:middle line:90% 00:32:28.933 --> 00:32:30.350 align:middle line:84% So we're going to use the method-- 00:32:30.350 --> 00:32:31.220 align:middle line:90% anyone remember? 00:32:31.220 --> 00:32:34.347 align:middle line:90% What method are we going to use? 00:32:34.347 --> 00:32:35.055 align:middle line:90% Is it from multi? 00:32:35.055 --> 00:32:38.930 align:middle line:90% 00:32:38.930 --> 00:32:42.160 align:middle line:90% French name. 00:32:42.160 --> 00:32:44.025 align:middle line:84% STUDENT: The Lagrange multiplier? 00:32:44.025 --> 00:32:46.072 align:middle line:90% 00:32:46.072 --> 00:32:47.030 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:32:47.030 --> 00:32:47.570 align:middle line:90% Thank you. 00:32:47.570 --> 00:32:51.012 align:middle line:90% 00:32:51.012 --> 00:32:53.470 align:middle line:84% We're going to use [INAUDIBLE] of the Lagrange multipliers. 00:32:53.470 --> 00:32:55.230 align:middle line:84% So in your calculus textbook, you 00:32:55.230 --> 00:32:56.710 align:middle line:84% would have used the del operator. 00:32:56.710 --> 00:32:59.650 align:middle line:84% I'll just write that here just for familiarity, 00:32:59.650 --> 00:33:01.662 align:middle line:84% but then we'll switch back to our operator. 00:33:01.662 --> 00:33:02.620 align:middle line:90% And so what do we have? 00:33:02.620 --> 00:33:04.510 align:middle line:84% Del, the thing we want to optimize, 00:33:04.510 --> 00:33:07.450 align:middle line:84% plus del, the things which are conserved. 00:33:07.450 --> 00:33:09.880 align:middle line:84% And we have Lagrange multipliers. 00:33:09.880 --> 00:33:12.850 align:middle line:84% So we have del n total and del u, 00:33:12.850 --> 00:33:15.310 align:middle line:84% and this whole thing is going to be 0. 00:33:15.310 --> 00:33:26.860 align:middle line:84% This is written as in your calc textbook with the del operator, 00:33:26.860 --> 00:33:32.860 align:middle line:84% but we're going to write this way, ds plus alpha dn 00:33:32.860 --> 00:33:38.590 align:middle line:90% total plus beta du equals 0. 00:33:38.590 --> 00:33:41.530 align:middle line:84% And alpha and beta are Lagrange multipliers. 00:33:41.530 --> 00:33:53.680 align:middle line:90% 00:33:53.680 --> 00:33:55.660 align:middle line:84% I was talking to my wife about this actually. 00:33:55.660 --> 00:33:57.400 align:middle line:90% She teaches college math. 00:33:57.400 --> 00:34:02.080 align:middle line:84% And it does seem this is the most important application 00:34:02.080 --> 00:34:10.300 align:middle line:84% of calculus outside of some specialty areas, at least 00:34:10.300 --> 00:34:11.560 align:middle line:90% the most widespread. 00:34:11.560 --> 00:34:13.630 align:middle line:90% Anyway. 00:34:13.630 --> 00:34:15.550 align:middle line:84% So we're going to substitute our expressions. 00:34:15.550 --> 00:34:24.540 align:middle line:90% 00:34:24.540 --> 00:34:34.639 align:middle line:84% We have expressions for ds and du and dn total 00:34:34.639 --> 00:34:35.730 align:middle line:90% and collect terms. 00:34:35.730 --> 00:34:36.980 align:middle line:90% That's what we're going to do. 00:34:36.980 --> 00:34:40.770 align:middle line:90% 00:34:40.770 --> 00:34:42.210 align:middle line:84% So we substitute our expressions, 00:34:42.210 --> 00:34:48.520 align:middle line:84% and we collect terms, and we get the following, sum over states, 00:34:48.520 --> 00:34:58.600 align:middle line:84% we're going to have minus k sub beta log n of i over n total 00:34:58.600 --> 00:35:10.300 align:middle line:84% plus alpha plus beta e of i, dn of i equals 0. 00:35:10.300 --> 00:35:14.520 align:middle line:90% So what is this functional form? 00:35:14.520 --> 00:35:15.270 align:middle line:90% What is this form? 00:35:15.270 --> 00:35:18.120 align:middle line:84% This is just like we did two months ago, 00:35:18.120 --> 00:35:21.660 align:middle line:84% starting about two months ago, for the case of unary systems. 00:35:21.660 --> 00:35:38.250 align:middle line:84% We have here a set of unconstrained independent 00:35:38.250 --> 00:35:40.060 align:middle line:90% variables. 00:35:40.060 --> 00:35:42.450 align:middle line:84% And so we want this whole thing to be zero. 00:35:42.450 --> 00:35:44.640 align:middle line:84% How do we ensure that this whole thing is zero? 00:35:44.640 --> 00:35:53.420 align:middle line:90% 00:35:53.420 --> 00:35:55.520 align:middle line:84% Remember these differential forms? 00:35:55.520 --> 00:35:58.850 align:middle line:84% What did we call the pre factor in front of the differential 00:35:58.850 --> 00:36:00.470 align:middle line:90% of the independent variables? 00:36:00.470 --> 00:36:03.710 align:middle line:90% 00:36:03.710 --> 00:36:06.028 align:middle line:84% STUDENT: You just said the coefficients equal zero. 00:36:06.028 --> 00:36:07.570 align:middle line:84% RAFAEL JARAMILLO: Coefficient, right? 00:36:07.570 --> 00:36:08.440 align:middle line:90% Exactly. 00:36:08.440 --> 00:36:22.260 align:middle line:84% We're going to set each coefficient to zero. 00:36:22.260 --> 00:36:36.490 align:middle line:84% So let's do that, minus kb log n of i over n total plus alpha, 00:36:36.490 --> 00:36:40.270 align:middle line:90% plus beta e of i equals 0. 00:36:40.270 --> 00:36:50.430 align:middle line:84% I'm going to rearrange n of i over n total equals e to alpha 00:36:50.430 --> 00:36:59.960 align:middle line:90% over kb, e beta e of i over kb. 00:36:59.960 --> 00:37:11.000 align:middle line:84% And this is true for each state I equals 1, 2, through, say, r. 00:37:11.000 --> 00:37:12.590 align:middle line:84% So we're not there yet, but we just 00:37:12.590 --> 00:37:14.298 align:middle line:84% had something really important to happen. 00:37:14.298 --> 00:37:16.690 align:middle line:90% 00:37:16.690 --> 00:37:19.990 align:middle line:90% This is a distribution function. 00:37:19.990 --> 00:37:20.750 align:middle line:90% All right. 00:37:20.750 --> 00:37:27.580 align:middle line:84% This is describing the occupancy of state I. 00:37:27.580 --> 00:37:28.960 align:middle line:90% And it's a fractional occupancy. 00:37:28.960 --> 00:37:31.750 align:middle line:84% It's n of i over n of total, so the fraction 00:37:31.750 --> 00:37:33.880 align:middle line:84% of particles that are in state I. 00:37:33.880 --> 00:37:40.610 align:middle line:84% And it's exponentially dependent on the energy of state I. 00:37:40.610 --> 00:37:41.110 align:middle line:90% All right. 00:37:41.110 --> 00:37:43.840 align:middle line:84% So I want you to notice two things here. 00:37:43.840 --> 00:37:46.510 align:middle line:90% I'll just repeat what I said. 00:37:46.510 --> 00:37:48.370 align:middle line:84% That this is a distribution function. 00:37:48.370 --> 00:37:51.930 align:middle line:90% 00:37:51.930 --> 00:37:55.670 align:middle line:90% That's a very useful thing. 00:37:55.670 --> 00:38:09.053 align:middle line:84% That's a distribution function, and it's exponential in e of i. 00:38:09.053 --> 00:38:11.470 align:middle line:84% But we haven't finished yet because we have these Lagrange 00:38:11.470 --> 00:38:11.950 align:middle line:90% multipliers. 00:38:11.950 --> 00:38:13.450 align:middle line:84% So we don't know what those are yet. 00:38:13.450 --> 00:38:15.710 align:middle line:84% So we're going to determine one of them now, 00:38:15.710 --> 00:38:19.780 align:middle line:84% and we'll determine the other one on Wednesday. 00:38:19.780 --> 00:38:29.720 align:middle line:84% So we're going to determine alpha by normalization. 00:38:29.720 --> 00:38:30.800 align:middle line:90% What do I mean by that? 00:38:30.800 --> 00:38:34.100 align:middle line:90% 00:38:34.100 --> 00:38:37.960 align:middle line:84% What I mean by that is the sum over all of i, 00:38:37.960 --> 00:38:40.030 align:middle line:90% n of i over n total-- 00:38:40.030 --> 00:38:43.735 align:middle line:84% what is this sum, sum over all of i, n of I over n total? 00:38:43.735 --> 00:38:46.250 align:middle line:90% 00:38:46.250 --> 00:38:47.240 align:middle line:90% STUDENT: Just one. 00:38:47.240 --> 00:38:49.690 align:middle line:84% RAFAEL JARAMILLO: Yeah, it's one. 00:38:49.690 --> 00:38:50.190 align:middle line:90% It's one. 00:38:50.190 --> 00:38:55.170 align:middle line:90% 00:38:55.170 --> 00:38:58.920 align:middle line:84% So if we set that equal to 1, we get 00:38:58.920 --> 00:39:03.890 align:middle line:84% the following, e to the alpha over Boltzmann's 00:39:03.890 --> 00:39:19.495 align:middle line:84% constant equals 1 over sum over i, e beta, epsilon i over kp. 00:39:19.495 --> 00:39:21.245 align:middle line:84% And we're going to give this thing a name. 00:39:21.245 --> 00:39:24.948 align:middle line:90% 00:39:24.948 --> 00:39:26.865 align:middle line:84% We're going to call it the partition function. 00:39:26.865 --> 00:39:32.880 align:middle line:90% 00:39:32.880 --> 00:39:38.410 align:middle line:84% The partition function Q is the sum over all possible states 00:39:38.410 --> 00:39:45.810 align:middle line:90% e beta e of i over k Boltzmann. 00:39:45.810 --> 00:39:47.790 align:middle line:90% So for now, it's just a name. 00:39:47.790 --> 00:39:48.820 align:middle line:90% What does that mean? 00:39:48.820 --> 00:39:53.190 align:middle line:90% It normalizes the distribution. 00:39:53.190 --> 00:39:54.720 align:middle line:84% Why is it called partition function? 00:39:54.720 --> 00:39:56.970 align:middle line:84% It describes all the different ways 00:39:56.970 --> 00:39:59.950 align:middle line:84% that the energy can be partitioned in the system. 00:39:59.950 --> 00:40:01.650 align:middle line:84% So it's a sum over all the states, 00:40:01.650 --> 00:40:04.810 align:middle line:84% somehow characterizing the system 00:40:04.810 --> 00:40:07.810 align:middle line:84% and all the ways energy can be partitioned in their. 00:40:07.810 --> 00:40:21.480 align:middle line:84% Partition function normalizes the distribution function. 00:40:21.480 --> 00:40:28.290 align:middle line:90% 00:40:28.290 --> 00:40:34.760 align:middle line:84% So this distribution function, n of i over n total 00:40:34.760 --> 00:40:46.470 align:middle line:84% is equal eb beta epsilon i over k beta divided by Q. 00:40:46.470 --> 00:40:51.150 align:middle line:84% So it looks like I end it about five minutes early, but better 00:40:51.150 --> 00:40:54.810 align:middle line:84% that than rush through the derivation 00:40:54.810 --> 00:40:57.430 align:middle line:90% of a determination of beta. 00:40:57.430 --> 00:41:00.840 align:middle line:84% So I'm going to stop now and take questions. 00:41:00.840 --> 00:41:06.130 align:middle line:90% 00:41:06.130 --> 00:41:08.520 align:middle line:84% STUDENT: Could you explain the Lagrange multipliers 00:41:08.520 --> 00:41:11.114 align:middle line:90% a little bit? 00:41:11.114 --> 00:41:12.012 align:middle line:90% I can't find it. 00:41:12.012 --> 00:41:12.970 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:41:12.970 --> 00:41:20.170 align:middle line:90% 00:41:20.170 --> 00:41:26.460 align:middle line:90% So you want to subject-- 00:41:26.460 --> 00:41:31.360 align:middle line:84% you want this to be 0 subject to the constraint that this is 0 00:41:31.360 --> 00:41:32.800 align:middle line:90% and also this is 0. 00:41:32.800 --> 00:41:35.610 align:middle line:90% 00:41:35.610 --> 00:41:39.720 align:middle line:84% And you know these are 0 because you're 00:41:39.720 --> 00:41:41.710 align:middle line:90% applying those constraints. 00:41:41.710 --> 00:41:45.460 align:middle line:84% And so you can add these to the equation of delta 00:41:45.460 --> 00:41:47.830 align:middle line:84% S equals 0 without fundamentally changing the equation. 00:41:47.830 --> 00:41:50.540 align:middle line:90% 00:41:50.540 --> 00:41:55.130 align:middle line:84% And then alpha and beta are not necessary mathematically, 00:41:55.130 --> 00:41:57.110 align:middle line:84% but they generalize the situation. 00:41:57.110 --> 00:41:59.640 align:middle line:90% 00:41:59.640 --> 00:42:02.360 align:middle line:84% And so then you can effectively relax-- 00:42:02.360 --> 00:42:05.090 align:middle line:84% you can relax your constraints here, 00:42:05.090 --> 00:42:10.340 align:middle line:84% and the overall constraint is maintained by alpha and beta. 00:42:10.340 --> 00:42:13.190 align:middle line:84% So your method of Lagrange multipliers 00:42:13.190 --> 00:42:16.065 align:middle line:90% is only valid with-- 00:42:16.065 --> 00:42:18.530 align:middle line:84% the equations that result from this 00:42:18.530 --> 00:42:22.100 align:middle line:84% appear as if they might be valid even when 00:42:22.100 --> 00:42:26.450 align:middle line:84% the constraints on total number of particles and energy relax, 00:42:26.450 --> 00:42:27.320 align:middle line:90% but they're not. 00:42:27.320 --> 00:42:29.720 align:middle line:90% You have to remember that. 00:42:29.720 --> 00:42:34.520 align:middle line:84% The equations that we get from this method 00:42:34.520 --> 00:42:37.580 align:middle line:84% are only applicable when u of t and n are fixed. 00:42:37.580 --> 00:42:42.650 align:middle line:90% 00:42:42.650 --> 00:42:45.640 align:middle line:84% So that's the concept point here. 00:42:45.640 --> 00:42:51.450 align:middle line:84% And so when we derive a distribution function, 00:42:51.450 --> 00:42:53.070 align:middle line:84% we have this distribution function, 00:42:53.070 --> 00:42:54.970 align:middle line:90% or we have this form here. 00:42:54.970 --> 00:42:58.320 align:middle line:84% And this, of course, also relates to the Arrhenius rate 00:42:58.320 --> 00:42:59.470 align:middle line:90% law. 00:42:59.470 --> 00:43:00.555 align:middle line:90% So we're getting there. 00:43:00.555 --> 00:43:06.610 align:middle line:84% And people will see this distribution function 00:43:06.610 --> 00:43:11.340 align:middle line:84% so often in natural sciences that it's 00:43:11.340 --> 00:43:15.570 align:middle line:84% important to remember it's not always true. 00:43:15.570 --> 00:43:19.740 align:middle line:84% It is very specifically true that it is the distribution 00:43:19.740 --> 00:43:24.940 align:middle line:84% function that maxima optimizes entropy under this condition. 00:43:24.940 --> 00:43:28.840 align:middle line:84% So I think that's the concept here. 00:43:28.840 --> 00:43:31.798 align:middle line:90% So don't apply it willy-nilly. 00:43:31.798 --> 00:43:32.590 align:middle line:90% STUDENT: Thank you. 00:43:32.590 --> 00:43:33.440 align:middle line:84% RAFAEL JARAMILLO: But you don't have to worry. 00:43:33.440 --> 00:43:35.110 align:middle line:84% This isn't a calculus class in the sense 00:43:35.110 --> 00:43:37.900 align:middle line:84% that I'm going to ask you to derive this or even repeat 00:43:37.900 --> 00:43:39.648 align:middle line:84% this derivation, but I do want you 00:43:39.648 --> 00:43:41.440 align:middle line:84% to know where this comes from, because this 00:43:41.440 --> 00:43:43.270 align:middle line:84% is going to start becoming second nature, 00:43:43.270 --> 00:43:45.100 align:middle line:84% not necessarily in the next week and a 1/2. 00:43:45.100 --> 00:43:46.510 align:middle line:84% But this distribution function is 00:43:46.510 --> 00:43:48.655 align:middle line:84% going to be so familiar to you by the time you're 00:43:48.655 --> 00:43:51.190 align:middle line:84% a fourth year student in material science. 00:43:51.190 --> 00:43:53.560 align:middle line:84% It's good to remember that it comes from somewhere, 00:43:53.560 --> 00:43:56.910 align:middle line:90% and it's subject to assumptions. 00:43:56.910 --> 00:43:59.000 align:middle line:90%