WEBVTT 00:00:00.000 --> 00:00:12.000 align:middle line:90% 00:00:12.000 --> 00:00:13.000 align:middle line:90% RAFAEL JARAMILLO: Right. 00:00:13.000 --> 00:00:14.417 align:middle line:84% So what we're going to do today is 00:00:14.417 --> 00:00:16.390 align:middle line:84% we're going to finish our coverage 00:00:16.390 --> 00:00:22.210 align:middle line:84% of statistical thermodynamics, and then we're 00:00:22.210 --> 00:00:26.410 align:middle line:84% going to do a couple lectures, starting Monday. 00:00:26.410 --> 00:00:27.310 align:middle line:90% Friday's a holiday. 00:00:27.310 --> 00:00:30.640 align:middle line:84% So we'll do a couple lectures, starting Monday 00:00:30.640 --> 00:00:37.680 align:middle line:84% on reacting systems, and that includes oxidation processes. 00:00:37.680 --> 00:00:41.597 align:middle line:84% And that's mainly to have chapter 11, and then 00:00:41.597 --> 00:00:42.180 align:middle line:90% that'll be it. 00:00:42.180 --> 00:00:43.555 align:middle line:84% Then we're going to have one more 00:00:43.555 --> 00:00:46.680 align:middle line:84% hour on social and personal, then 00:00:46.680 --> 00:00:48.510 align:middle line:84% we're going to have a game show, which I 00:00:48.510 --> 00:00:49.620 align:middle line:90% don't know how to run a resume. 00:00:49.620 --> 00:00:50.537 align:middle line:90% We're going to try it. 00:00:50.537 --> 00:00:58.260 align:middle line:90% 00:00:58.260 --> 00:01:01.680 align:middle line:84% And then the last exam, which is noncumulative. 00:01:01.680 --> 00:01:04.260 align:middle line:84% So let's pick up where we left off, 00:01:04.260 --> 00:01:11.270 align:middle line:84% max entropy and the Boltzmann distribution. 00:01:11.270 --> 00:01:18.000 align:middle line:90% 00:01:18.000 --> 00:01:23.040 align:middle line:84% And this is continued, continued from the last lecture. 00:01:23.040 --> 00:01:25.130 align:middle line:84% So I don't like to do this, where 00:01:25.130 --> 00:01:27.930 align:middle line:84% I continue a derivation of something across two lectures, 00:01:27.930 --> 00:01:29.390 align:middle line:90% but that's how I broke. 00:01:29.390 --> 00:01:32.530 align:middle line:90% 00:01:32.530 --> 00:01:35.590 align:middle line:84% So from last time, just at a high level, 00:01:35.590 --> 00:01:43.190 align:middle line:84% we decided that we wanted to optimize entropy 00:01:43.190 --> 00:01:44.420 align:middle line:90% for an isolated system. 00:01:44.420 --> 00:01:49.170 align:middle line:90% 00:01:49.170 --> 00:01:51.770 align:middle line:84% And we had Boltzmann entropy formula, 00:01:51.770 --> 00:01:54.040 align:middle line:90% so we could do some math. 00:01:54.040 --> 00:02:01.510 align:middle line:84% And we could write down the S equals 0 subject 00:02:01.510 --> 00:02:02.470 align:middle line:90% to constraints. 00:02:02.470 --> 00:02:05.670 align:middle line:90% 00:02:05.670 --> 00:02:09.630 align:middle line:84% And the constraints being conservation of energy 00:02:09.630 --> 00:02:11.460 align:middle line:90% and conservation of mass. 00:02:11.460 --> 00:02:15.830 align:middle line:90% 00:02:15.830 --> 00:02:18.470 align:middle line:84% And we're going to do this using the method of Lagrange 00:02:18.470 --> 00:02:19.520 align:middle line:90% multipliers. 00:02:19.520 --> 00:02:29.810 align:middle line:90% 00:02:29.810 --> 00:02:34.120 align:middle line:84% And we had two of them because we had two constraints, alpha 00:02:34.120 --> 00:02:35.290 align:middle line:90% and beta. 00:02:35.290 --> 00:02:37.030 align:middle line:84% And we ended up with an expression 00:02:37.030 --> 00:02:45.430 align:middle line:84% like this, some stuff, so coefficient, 00:02:45.430 --> 00:02:47.020 align:middle line:90% independent variables. 00:02:47.020 --> 00:02:50.950 align:middle line:84% And in order to get our max entropy condition, 00:02:50.950 --> 00:02:52.930 align:middle line:84% I should say optimize entropy, we 00:02:52.930 --> 00:02:54.955 align:middle line:84% have to set these coefficients to zero. 00:02:54.955 --> 00:02:59.370 align:middle line:90% 00:02:59.370 --> 00:03:02.490 align:middle line:84% So it was a very familiar thermodynamic thing 00:03:02.490 --> 00:03:06.760 align:middle line:90% that we did, and what? 00:03:06.760 --> 00:03:14.070 align:middle line:84% We determined the first multiplier by normalization. 00:03:14.070 --> 00:03:16.810 align:middle line:90% 00:03:16.810 --> 00:03:19.410 align:middle line:90% So this is where we left off. 00:03:19.410 --> 00:03:21.910 align:middle line:84% Where we left off was the following. 00:03:21.910 --> 00:03:24.150 align:middle line:90% We had ni over n total. 00:03:24.150 --> 00:03:27.510 align:middle line:84% So this is a distribution function 00:03:27.510 --> 00:03:30.690 align:middle line:84% of fractional distributions of particles in state 00:03:30.690 --> 00:03:35.730 align:middle line:84% I normalized by the total number of particles. 00:03:35.730 --> 00:03:40.590 align:middle line:84% And it was like this, this multiplayer beta 00:03:40.590 --> 00:03:44.940 align:middle line:84% as yet undetermined, energy of state I normalized 00:03:44.940 --> 00:03:47.600 align:middle line:90% by Boltzmann's constant. 00:03:47.600 --> 00:03:50.180 align:middle line:84% And that whole thing normalized by Q, which 00:03:50.180 --> 00:03:51.990 align:middle line:90% was the partition function. 00:03:51.990 --> 00:03:54.560 align:middle line:84% And this was the sum of all possible states 00:03:54.560 --> 00:03:59.280 align:middle line:84% e to the beta epsilon i over Boltzmann's constant. 00:03:59.280 --> 00:04:01.085 align:middle line:84% And so this is where we left things. 00:04:01.085 --> 00:04:04.010 align:middle line:90% 00:04:04.010 --> 00:04:07.010 align:middle line:84% So now what we're going to do is we're going to determine beta. 00:04:07.010 --> 00:04:11.900 align:middle line:90% 00:04:11.900 --> 00:04:19.649 align:middle line:84% We're going to determine beta by considering 00:04:19.649 --> 00:04:31.870 align:middle line:84% a microscopic, reversible process where 00:04:31.870 --> 00:04:35.047 align:middle line:84% some of the particles change their shape. 00:04:35.047 --> 00:04:36.130 align:middle line:90% So we're going to do this. 00:04:36.130 --> 00:04:41.420 align:middle line:84% We're going to analyze ds for a process of some state changes. 00:04:41.420 --> 00:04:44.170 align:middle line:84% And I'm going to just go ahead and write down 00:04:44.170 --> 00:04:47.740 align:middle line:84% the equations that we've been using and developing for this. 00:04:47.740 --> 00:04:51.505 align:middle line:84% ds equals log n of i over n total. 00:04:51.505 --> 00:04:55.400 align:middle line:90% 00:04:55.400 --> 00:05:01.947 align:middle line:84% dn of i, you can pull that from the last two lectures. 00:05:01.947 --> 00:05:03.530 align:middle line:84% But now we actually have an expression 00:05:03.530 --> 00:05:08.310 align:middle line:84% for this distribution, so we're going to use that expression, 00:05:08.310 --> 00:05:13.165 align:middle line:90% kb sum over i. 00:05:13.165 --> 00:05:17.100 align:middle line:84% The distribution is e to the beta epsilon 00:05:17.100 --> 00:05:25.070 align:middle line:84% i over Boltzmann constant over Q, 00:05:25.070 --> 00:05:27.540 align:middle line:84% then we have some logs of exponentials and things. 00:05:27.540 --> 00:05:29.300 align:middle line:84% So we can simplify this a little bit. 00:05:29.300 --> 00:05:42.920 align:middle line:84% And this equals minus beta sum over i epsilon i dn of i 00:05:42.920 --> 00:05:54.930 align:middle line:84% plus k beta log of Q sum over i dn of i. 00:05:54.930 --> 00:06:11.380 align:middle line:84% And taking our definitions, this equals minus beta du plus kb 00:06:11.380 --> 00:06:15.865 align:middle line:90% log of Q dn total. 00:06:15.865 --> 00:06:19.210 align:middle line:90% 00:06:19.210 --> 00:06:27.830 align:middle line:84% So first thing to note, this is zero by construction. 00:06:27.830 --> 00:06:34.270 align:middle line:84% This is the Lagrange multiplier working, because we wanted ds 00:06:34.270 --> 00:06:40.580 align:middle line:84% to be 0 under conditions where du was 0 and dn total were 0. 00:06:40.580 --> 00:06:49.175 align:middle line:84% And we see that for this distribution, it works out. 00:06:49.175 --> 00:06:50.870 align:middle line:90% So Lagrange multiplier worked. 00:06:50.870 --> 00:06:53.700 align:middle line:90% The method worked. 00:06:53.700 --> 00:06:56.270 align:middle line:84% So I'm going to say 0 by construction. 00:06:56.270 --> 00:07:04.030 align:middle line:90% 00:07:04.030 --> 00:07:07.060 align:middle line:84% I really like this orange pen, but life is 00:07:07.060 --> 00:07:10.605 align:middle line:90% too short for fading sharpies. 00:07:10.605 --> 00:07:14.170 align:middle line:90% 00:07:14.170 --> 00:07:15.650 align:middle line:84% So by construction, this is zero. 00:07:15.650 --> 00:07:17.890 align:middle line:84% If you'd like, you can convince yourself 00:07:17.890 --> 00:07:20.847 align:middle line:84% that if you have a different distribution function here, 00:07:20.847 --> 00:07:22.930 align:middle line:84% if you have a different distribution function, not 00:07:22.930 --> 00:07:25.270 align:middle line:84% this distribution, let's say a flat distribution 00:07:25.270 --> 00:07:27.130 align:middle line:84% function, a distribution function that 00:07:27.130 --> 00:07:29.770 align:middle line:84% doesn't depend on energy or has some other functional 00:07:29.770 --> 00:07:32.703 align:middle line:84% dependence of energy, I don't know what polynomials, 00:07:32.703 --> 00:07:34.120 align:middle line:84% you can put anything in there, you 00:07:34.120 --> 00:07:40.200 align:middle line:84% can show that that in that case, ds will not be 0. 00:07:40.200 --> 00:07:43.650 align:middle line:84% So again, Lagrange multiplier method works. 00:07:43.650 --> 00:07:45.180 align:middle line:90% But here's the takeaway. 00:07:45.180 --> 00:07:52.070 align:middle line:84% We want to compare this to what we have derived 00:07:52.070 --> 00:07:57.120 align:middle line:84% two months ago from the combined statement, which 00:07:57.120 --> 00:08:09.600 align:middle line:84% is that ds equals 1 over t du plus p over t dv 00:08:09.600 --> 00:08:15.200 align:middle line:90% minus mu over t dn. 00:08:15.200 --> 00:08:18.080 align:middle line:84% And this is really the important thing here. 00:08:18.080 --> 00:08:24.230 align:middle line:84% We are comparing the coefficients. 00:08:24.230 --> 00:08:26.360 align:middle line:84% We're comparing the coefficients. 00:08:26.360 --> 00:08:29.480 align:middle line:84% And if you remember, this is almost 00:08:29.480 --> 00:08:31.640 align:middle line:90% like general strategy stuff. 00:08:31.640 --> 00:08:34.890 align:middle line:90% 00:08:34.890 --> 00:08:40.260 align:middle line:84% If you compare coefficients, you can buy inspection, 00:08:40.260 --> 00:08:43.030 align:middle line:90% set like coefficients to like. 00:08:43.030 --> 00:08:52.590 align:middle line:84% So what is ds, du at fixed volume and fixed particle 00:08:52.590 --> 00:08:53.610 align:middle line:90% number? 00:08:53.610 --> 00:08:56.890 align:middle line:84% Well, from our statistical method, 00:08:56.890 --> 00:09:01.440 align:middle line:84% that's minus beta, but from the classical method, 00:09:01.440 --> 00:09:03.660 align:middle line:90% that's 1 over temperature. 00:09:03.660 --> 00:09:09.120 align:middle line:84% And so we can simply conclude that beta equals minus 1 00:09:09.120 --> 00:09:10.377 align:middle line:90% over temperature. 00:09:10.377 --> 00:09:16.590 align:middle line:90% 00:09:16.590 --> 00:09:19.500 align:middle line:84% So now with that identified, we have the Boltzmann 00:09:19.500 --> 00:09:20.090 align:middle line:90% distribution. 00:09:20.090 --> 00:09:23.820 align:middle line:90% Oh, there's more paper. 00:09:23.820 --> 00:09:30.640 align:middle line:84% So with beta equals minus 1 over temperature, 00:09:30.640 --> 00:09:36.070 align:middle line:84% we have the Boltzmann distribution. 00:09:36.070 --> 00:09:45.470 align:middle line:90% 00:09:45.470 --> 00:09:49.600 align:middle line:84% Number of particles in state I normalized 00:09:49.600 --> 00:09:52.690 align:middle line:84% by the total number of particles equals 00:09:52.690 --> 00:10:03.060 align:middle line:84% e to the minus Ei over kbT normalized by the partition 00:10:03.060 --> 00:10:05.310 align:middle line:84% function, where the partition function 00:10:05.310 --> 00:10:10.905 align:middle line:84% equals the sum over all states e to the minus Ei over kbT. 00:10:10.905 --> 00:10:15.610 align:middle line:90% 00:10:15.610 --> 00:10:22.700 align:middle line:84% And this generally has a falling exponential form 00:10:22.700 --> 00:10:25.290 align:middle line:90% with state energy. 00:10:25.290 --> 00:10:31.280 align:middle line:84% So state e of i, occupation n of i, 00:10:31.280 --> 00:10:33.365 align:middle line:84% all right, so falling exponential function. 00:10:33.365 --> 00:10:38.060 align:middle line:90% 00:10:38.060 --> 00:10:46.230 align:middle line:84% In words, this is the distribution 00:10:46.230 --> 00:11:05.510 align:middle line:84% that maximizes entropy for an isolated system 00:11:05.510 --> 00:11:23.160 align:middle line:84% with n total particles distributed over states 00:11:23.160 --> 00:11:32.820 align:middle line:84% with fixed energy levels, so unchanging energy levels, 00:11:32.820 --> 00:11:42.390 align:middle line:84% according to that set of occupation numbers. 00:11:42.390 --> 00:11:46.650 align:middle line:84% And what is the state of maximum entropy for an isolated system? 00:11:46.650 --> 00:11:49.248 align:middle line:90% We have a name for that. 00:11:49.248 --> 00:11:51.540 align:middle line:84% What's the state of max entropy for an isolated system? 00:11:51.540 --> 00:12:00.280 align:middle line:90% 00:12:00.280 --> 00:12:04.060 align:middle line:84% Think way back to late February even. 00:12:04.060 --> 00:12:10.170 align:middle line:90% 00:12:10.170 --> 00:12:11.632 align:middle line:90% STUDENT: Maybe equilibrium? 00:12:11.632 --> 00:12:13.132 align:middle line:84% RAFAEL JARAMILLO: Yeah, equilibrium. 00:12:13.132 --> 00:12:16.740 align:middle line:90% 00:12:16.740 --> 00:12:20.970 align:middle line:84% So this is an equilibrium distribution. 00:12:20.970 --> 00:12:23.540 align:middle line:84% So we have an equilibrium distribution 00:12:23.540 --> 00:12:25.970 align:middle line:84% of single-particle energy levels, 00:12:25.970 --> 00:12:29.030 align:middle line:90% or single-particle energies. 00:12:29.030 --> 00:12:32.630 align:middle line:84% And again, you knew this already, 00:12:32.630 --> 00:12:36.110 align:middle line:84% but equilibrium does not mean everything 00:12:36.110 --> 00:12:37.580 align:middle line:90% is at the lowest energy. 00:12:37.580 --> 00:12:39.942 align:middle line:84% All semester, we've been exploring that consequence. 00:12:39.942 --> 00:12:47.440 align:middle line:84% So if you were at zero Kelvin, if you're at zero Kelvin, 00:12:47.440 --> 00:12:50.430 align:middle line:84% or you're a physicist, which means you have a frigid heart, 00:12:50.430 --> 00:12:53.260 align:middle line:84% you're zero Kelvin, this is your distribution function. 00:12:53.260 --> 00:12:58.100 align:middle line:84% Everything's piled up, has zero energy. 00:12:58.100 --> 00:13:02.978 align:middle line:84% But in any real temperature, you get energetic particles, 00:13:02.978 --> 00:13:04.895 align:middle line:84% and you get them with a distribution function. 00:13:04.895 --> 00:13:10.230 align:middle line:90% 00:13:10.230 --> 00:13:11.640 align:middle line:84% So a couple of observations here. 00:13:11.640 --> 00:13:15.930 align:middle line:84% The first observation is maximum distribution-- 00:13:15.930 --> 00:13:21.440 align:middle line:90% 00:13:21.440 --> 00:13:32.040 align:middle line:84% sorry, said maximum-- Boltzmann distribution and likelihood. 00:13:32.040 --> 00:13:36.270 align:middle line:90% 00:13:36.270 --> 00:13:41.220 align:middle line:84% Again likelihood, this is a stealth class on statistics. 00:13:41.220 --> 00:13:43.050 align:middle line:84% That's what statistical thermodynamics is. 00:13:43.050 --> 00:13:45.000 align:middle line:84% It's like a stealth class on statistics. 00:13:45.000 --> 00:13:57.360 align:middle line:84% So we'll ask, how much less likely 00:13:57.360 --> 00:14:17.260 align:middle line:84% is it to find a particle in state with energy, let's say, 00:14:17.260 --> 00:14:18.880 align:middle line:90% Em-- 00:14:18.880 --> 00:14:20.290 align:middle line:90% that's an ugly epsilon-- 00:14:20.290 --> 00:14:36.250 align:middle line:84% equals El plus delta e then in state with energy El? 00:14:36.250 --> 00:14:40.630 align:middle line:84% So that's a typical thing that you might need to analyze. 00:14:40.630 --> 00:14:43.560 align:middle line:90% So let's see. 00:14:43.560 --> 00:14:47.940 align:middle line:84% nm over nl, that's how you turn the word problem 00:14:47.940 --> 00:14:49.480 align:middle line:90% into a math problem. 00:14:49.480 --> 00:14:53.430 align:middle line:84% How much less likely is it to find this distribution 00:14:53.430 --> 00:14:55.480 align:middle line:90% and this distribution? 00:14:55.480 --> 00:14:57.090 align:middle line:84% Within the equilibrium distribution, 00:14:57.090 --> 00:14:59.370 align:middle line:84% how much smaller is this number than this number? 00:14:59.370 --> 00:15:05.240 align:middle line:84% And you'll find the partition functions cancel out, 00:15:05.240 --> 00:15:14.680 align:middle line:84% and you get the following, El plus delta e over kt over e 00:15:14.680 --> 00:15:19.950 align:middle line:90% to the minus el ovet kT. 00:15:19.950 --> 00:15:22.230 align:middle line:84% And so the e to the minus Els cancel, 00:15:22.230 --> 00:15:29.280 align:middle line:84% and you get e to the minus delta e over kT. 00:15:29.280 --> 00:15:34.880 align:middle line:90% 00:15:34.880 --> 00:15:35.380 align:middle line:90% Right. 00:15:35.380 --> 00:15:46.610 align:middle line:84% So this depends on delta epsilon over kT, clearly. 00:15:46.610 --> 00:16:07.980 align:middle line:84% It becomes less likely with increasing energy splitting 00:16:07.980 --> 00:16:20.012 align:middle line:84% delta e, and it becomes more likely with temperature. 00:16:20.012 --> 00:16:26.290 align:middle line:90% 00:16:26.290 --> 00:16:27.210 align:middle line:90% All right. 00:16:27.210 --> 00:16:30.270 align:middle line:84% So you're going to explore this a little bit on the p set 00:16:30.270 --> 00:16:32.652 align:middle line:84% and the context of semiconductors. 00:16:32.652 --> 00:16:36.100 align:middle line:90% 00:16:36.100 --> 00:16:42.110 align:middle line:84% Another point, the thermal energy. 00:16:42.110 --> 00:16:50.470 align:middle line:90% 00:16:50.470 --> 00:16:52.580 align:middle line:84% This quantity is really important. 00:16:52.580 --> 00:16:54.610 align:middle line:84% It shows up in the natural sciences 00:16:54.610 --> 00:16:57.860 align:middle line:90% over and over and over again. 00:16:57.860 --> 00:16:59.840 align:middle line:90% And we're starting to see why. 00:16:59.840 --> 00:17:02.750 align:middle line:84% The why is what we've just shown, 00:17:02.750 --> 00:17:05.240 align:middle line:84% but the implications are really broad. 00:17:05.240 --> 00:17:07.140 align:middle line:84% You're going to see this all over the place. 00:17:07.140 --> 00:17:09.440 align:middle line:84% So let's remind ourselves what Boltzmann constant is. 00:17:09.440 --> 00:17:11.780 align:middle line:90% It's R over Na-- 00:17:11.780 --> 00:17:12.980 align:middle line:90% oh, that's funny-- RNa. 00:17:12.980 --> 00:17:16.520 align:middle line:84% So it's ideal gas constant divided by Avogadro's number. 00:17:16.520 --> 00:17:18.060 align:middle line:90% We have that from before. 00:17:18.060 --> 00:17:20.270 align:middle line:90% And this is something-- 00:17:20.270 --> 00:17:27.060 align:middle line:84% let's see, 1.380e to the minus 23 joules per Kelvin. 00:17:27.060 --> 00:17:29.433 align:middle line:90% So it has units of entropy. 00:17:29.433 --> 00:17:31.100 align:middle line:84% But that's not a number that I remember. 00:17:31.100 --> 00:17:34.160 align:middle line:90% Here's the number I do remember. 00:17:34.160 --> 00:17:40.400 align:middle line:84% And if you work on the molecular systems, or semiconductor 00:17:40.400 --> 00:17:45.490 align:middle line:84% systems, pretty much-- well, gee, 00:17:45.490 --> 00:17:47.410 align:middle line:84% a lot of systems that we work on in DMSE. 00:17:47.410 --> 00:17:49.750 align:middle line:84% This is a useful number to have memorized, 00:17:49.750 --> 00:17:55.540 align:middle line:84% Boltzmann's constant and units of eV per coulomb, 8.617e-5. 00:17:55.540 --> 00:17:59.140 align:middle line:84% Why on earth would you memorize that? 00:17:59.140 --> 00:18:11.120 align:middle line:84% It sets the energy scale for likely fluctuations. 00:18:11.120 --> 00:18:16.480 align:middle line:90% 00:18:16.480 --> 00:18:20.110 align:middle line:84% And we have-- all these words have meaning, 00:18:20.110 --> 00:18:23.200 align:middle line:90% likely fluctuations. 00:18:23.200 --> 00:18:25.880 align:middle line:84% Those words carry a lot of weight. 00:18:25.880 --> 00:18:29.440 align:middle line:84% So fluctuation, single-particle fluctuations 00:18:29.440 --> 00:18:37.310 align:middle line:84% with energy on the order of KbT are they're likely. 00:18:37.310 --> 00:18:39.670 align:middle line:84% That's something you're going to find, 00:18:39.670 --> 00:18:43.750 align:middle line:84% naturally occurring, as opposed to fluctuations, 00:18:43.750 --> 00:18:47.960 align:middle line:84% which are much, much larger than the thermal energy. 00:18:47.960 --> 00:18:49.270 align:middle line:90% Those are unlikely. 00:18:49.270 --> 00:18:56.823 align:middle line:90% 00:18:56.823 --> 00:18:58.990 align:middle line:84% We have all sorts of language for this that comes up 00:18:58.990 --> 00:19:00.550 align:middle line:90% in all sorts of contexts. 00:19:00.550 --> 00:19:05.290 align:middle line:84% For example, these fluctuations might be called 00:19:05.290 --> 00:19:08.970 align:middle line:90% thermally activated processes. 00:19:08.970 --> 00:19:13.050 align:middle line:90% 00:19:13.050 --> 00:19:15.370 align:middle line:84% You need a little bit of thermal energy to activate it. 00:19:15.370 --> 00:19:18.470 align:middle line:90% 00:19:18.470 --> 00:19:18.970 align:middle line:90% All right. 00:19:18.970 --> 00:19:21.460 align:middle line:84% So not only is this a number you should know, 00:19:21.460 --> 00:19:25.370 align:middle line:84% but the thermal energy at room temperature 00:19:25.370 --> 00:19:27.758 align:middle line:90% is a number you should know. 00:19:27.758 --> 00:19:34.150 align:middle line:90% A 298 Kelvin KbT equals-- 00:19:34.150 --> 00:19:36.040 align:middle line:84% well, in joules, it's not a number 00:19:36.040 --> 00:19:39.730 align:middle line:84% that I know because it's kind of unwieldy, 00:19:39.730 --> 00:19:42.330 align:middle line:84% but in eV, it is a number I know. 00:19:42.330 --> 00:19:45.740 align:middle line:84% So that's the natural energy scale of molecular processes. 00:19:45.740 --> 00:19:54.380 align:middle line:84% So it's 0.0257 electron volts, or 25.7 millielectron volts. 00:19:54.380 --> 00:19:57.310 align:middle line:84% So approximately 25 millielectron volts 00:19:57.310 --> 00:19:59.830 align:middle line:84% is the thermal energy at room temperature. 00:19:59.830 --> 00:20:02.430 align:middle line:84% So if you have a molecular system or an atomic system 00:20:02.430 --> 00:20:06.030 align:middle line:84% or an electronic system, processes with that energy 00:20:06.030 --> 00:20:08.070 align:middle line:84% are likely to be happening spontaneously 00:20:08.070 --> 00:20:09.810 align:middle line:90% at ambient temperature. 00:20:09.810 --> 00:20:11.580 align:middle line:84% Processes with much, much higher energy 00:20:11.580 --> 00:20:13.770 align:middle line:84% are not likely to be happening spontaneously 00:20:13.770 --> 00:20:16.520 align:middle line:90% at ambient temperature. 00:20:16.520 --> 00:20:18.680 align:middle line:84% So you can see how powerful this is. 00:20:18.680 --> 00:20:26.120 align:middle line:84% For example, it's the foundation for the Arrhenius rate 00:20:26.120 --> 00:20:26.620 align:middle line:90% equation. 00:20:26.620 --> 00:20:33.740 align:middle line:90% 00:20:33.740 --> 00:20:35.530 align:middle line:90% The Arrhenius rate equation-- 00:20:35.530 --> 00:20:37.910 align:middle line:84% and I'm going to use the notation from a chemistry class 00:20:37.910 --> 00:20:38.750 align:middle line:90% here-- 00:20:38.750 --> 00:20:42.140 align:middle line:84% a rate k equals a pre factor times e 00:20:42.140 --> 00:20:46.820 align:middle line:84% to the minus activation energy divided by Boltzmann's 00:20:46.820 --> 00:20:49.415 align:middle line:90% constant times temperature. 00:20:49.415 --> 00:20:49.940 align:middle line:90% All right. 00:20:49.940 --> 00:20:54.350 align:middle line:90% Now, this is a rate. 00:20:54.350 --> 00:21:04.800 align:middle line:84% This is an activation energy, rate and activation energy. 00:21:04.800 --> 00:21:10.383 align:middle line:84% And we are going to say hello to 3091 here. 00:21:10.383 --> 00:21:11.800 align:middle line:84% Actually may have seen that there. 00:21:11.800 --> 00:21:13.590 align:middle line:84% So that's another context, where you're 00:21:13.590 --> 00:21:17.040 align:middle line:84% going to see these exponentials, say why, where. 00:21:17.040 --> 00:21:18.340 align:middle line:90% Where does that come from? 00:21:18.340 --> 00:21:19.770 align:middle line:84% It comes from this Boltzmann distribution 00:21:19.770 --> 00:21:20.645 align:middle line:90% that we just derived. 00:21:20.645 --> 00:21:27.435 align:middle line:90% 00:21:27.435 --> 00:21:27.935 align:middle line:90% More. 00:21:27.935 --> 00:21:30.480 align:middle line:90% 00:21:30.480 --> 00:21:33.660 align:middle line:84% Let's talk about-- and for one slide on this, a little dense, 00:21:33.660 --> 00:21:35.730 align:middle line:84% but this is to help you with later 00:21:35.730 --> 00:21:38.130 align:middle line:84% Let's talk about the Maxwell-Boltzmann distribution. 00:21:38.130 --> 00:21:50.710 align:middle line:90% 00:21:50.710 --> 00:21:56.140 align:middle line:84% And I posted the wiki entry for this on website. 00:21:56.140 --> 00:21:59.350 align:middle line:84% So it's a good article to read, and I 00:21:59.350 --> 00:22:01.490 align:middle line:84% think that the lab instructors refer to it as well. 00:22:01.490 --> 00:22:18.840 align:middle line:84% So we're going to consider particles with kinetic energy. 00:22:18.840 --> 00:22:26.320 align:middle line:84% Kinetic energy is momentum vector modulus squared 00:22:26.320 --> 00:22:27.610 align:middle line:90% divided by 2 times a mass. 00:22:27.610 --> 00:22:30.520 align:middle line:84% That's kinetic energy of a particle of mass. 00:22:30.520 --> 00:22:33.490 align:middle line:84% And we're in three space, so this 00:22:33.490 --> 00:22:42.990 align:middle line:84% is px squared plus py squared plus pz squared divided by 2n. 00:22:42.990 --> 00:22:45.840 align:middle line:84% So that's the energy per particle and state 00:22:45.840 --> 00:22:46.920 align:middle line:90% of momentum p. 00:22:46.920 --> 00:22:50.730 align:middle line:90% 00:22:50.730 --> 00:22:53.070 align:middle line:84% So what we want to do is we want to figure out 00:22:53.070 --> 00:22:56.880 align:middle line:84% the distribution of particles as a function of energy. 00:22:56.880 --> 00:23:01.550 align:middle line:90% 00:23:01.550 --> 00:23:16.880 align:middle line:84% So this distribution in energy, I use f of v for consistency 00:23:16.880 --> 00:23:19.100 align:middle line:90% with Wikipedia. 00:23:19.100 --> 00:23:30.280 align:middle line:90% It depends on both the Maxwell-- 00:23:30.280 --> 00:23:34.250 align:middle line:84% sorry-- the Boltzmann distribution-- 00:23:34.250 --> 00:23:37.040 align:middle line:84% I had an error in my notes there-- 00:23:37.040 --> 00:23:47.010 align:middle line:84% the Boston distribution n of 1 over n total 00:23:47.010 --> 00:23:55.100 align:middle line:84% equals Q to the minus 1 exponent. 00:23:55.100 --> 00:23:59.870 align:middle line:84% And it's going to be minus momentum of state I 00:23:59.870 --> 00:24:03.540 align:middle line:90% modulus squared over 2 MkT. 00:24:03.540 --> 00:24:07.070 align:middle line:90% 00:24:07.070 --> 00:24:09.450 align:middle line:84% So it depends on that, the Boltzmann distribution, 00:24:09.450 --> 00:24:19.990 align:middle line:84% and also it depends on the density of states in momentum. 00:24:19.990 --> 00:24:23.910 align:middle line:90% 00:24:23.910 --> 00:24:25.310 align:middle line:84% This concept of density of states 00:24:25.310 --> 00:24:28.310 align:middle line:84% is not something that we've covered in this class. 00:24:28.310 --> 00:24:32.780 align:middle line:84% If you are in 029, you've discussed this. 00:24:32.780 --> 00:24:35.670 align:middle line:84% You're going to see it in the lab, 00:24:35.670 --> 00:24:37.430 align:middle line:84% but this really is a preview of what 00:24:37.430 --> 00:24:43.820 align:middle line:84% comes in the fall when density of states becomes kind of-- 00:24:43.820 --> 00:24:46.010 align:middle line:84% will be second nature for you in the context 00:24:46.010 --> 00:24:47.190 align:middle line:90% of quantum mechanics. 00:24:47.190 --> 00:24:49.220 align:middle line:84% But density of states means, well, this 00:24:49.220 --> 00:24:51.530 align:middle line:84% is the Boltzmann distribution for a given state 00:24:51.530 --> 00:24:55.300 align:middle line:90% I with this energy. 00:24:55.300 --> 00:24:58.540 align:middle line:90% How many such states are there? 00:24:58.540 --> 00:25:01.960 align:middle line:90% How many such states are there? 00:25:01.960 --> 00:25:16.310 align:middle line:84% So 3d differential, 4 equals 4 pi p quantity squared dp. 00:25:16.310 --> 00:25:18.973 align:middle line:90% 00:25:18.973 --> 00:25:20.390 align:middle line:84% We get the 4 pi there because they 00:25:20.390 --> 00:25:23.480 align:middle line:84% integrated the spherical coordinates, 00:25:23.480 --> 00:25:25.310 align:middle line:84% 4 pi radians all around the sphere. 00:25:25.310 --> 00:25:35.940 align:middle line:84% And that equals 4 pi m root 2 mE dE. 00:25:35.940 --> 00:25:41.010 align:middle line:84% So this is a distribution of states in energy E. 00:25:41.010 --> 00:25:42.960 align:middle line:84% And you combine these two things, 00:25:42.960 --> 00:25:44.640 align:middle line:84% you combine the density of states 00:25:44.640 --> 00:25:46.710 align:middle line:84% and the Boltzmann distribution, and you 00:25:46.710 --> 00:25:49.890 align:middle line:84% get the Maxwell-Boltzmann distribution, distribution 00:25:49.890 --> 00:25:52.560 align:middle line:84% of energetic particles zipping around 00:25:52.560 --> 00:25:57.240 align:middle line:84% on the kinetic energy included as a function of energy. 00:25:57.240 --> 00:26:10.520 align:middle line:84% And it is 2 E over pi, 1 over kT to the 3/2, e to the minus 00:26:10.520 --> 00:26:11.945 align:middle line:90% e over kT. 00:26:11.945 --> 00:26:14.870 align:middle line:90% 00:26:14.870 --> 00:26:18.660 align:middle line:84% So this is the Maxwell-Boltzmann distribution. 00:26:18.660 --> 00:26:25.930 align:middle line:84% And if you plot for example particle 00:26:25.930 --> 00:26:37.480 align:middle line:84% speed over probability, here's a low temperature, 00:26:37.480 --> 00:26:43.710 align:middle line:84% you have something that's pretty sharply spiked here, 00:26:43.710 --> 00:26:52.952 align:middle line:84% intermediate temperature, and then an high temperature. 00:26:52.952 --> 00:26:59.850 align:middle line:90% 00:26:59.850 --> 00:27:03.680 align:middle line:84% This is a series of increasing temp. 00:27:03.680 --> 00:27:12.870 align:middle line:90% 00:27:12.870 --> 00:27:16.110 align:middle line:84% And so this has the feature that-- 00:27:16.110 --> 00:27:21.150 align:middle line:84% there are more particles that go faster at higher temperature. 00:27:21.150 --> 00:27:25.850 align:middle line:84% You can see that coming from this exponential. 00:27:25.850 --> 00:27:28.100 align:middle line:84% And there's also a vanishingly small number 00:27:28.100 --> 00:27:29.465 align:middle line:90% of particles at zero speed. 00:27:29.465 --> 00:27:32.810 align:middle line:90% 00:27:32.810 --> 00:27:38.260 align:middle line:84% And there's going to be some mean speed that 00:27:38.260 --> 00:27:40.090 align:middle line:90% characterizes the distribution. 00:27:40.090 --> 00:27:44.120 align:middle line:84% And that mean moves with temperature. 00:27:44.120 --> 00:27:45.980 align:middle line:84% So I haven't derived this in detail 00:27:45.980 --> 00:27:48.470 align:middle line:84% because we haven't spent time on density of states, 00:27:48.470 --> 00:27:52.070 align:middle line:84% but we're just exploring implications of that Boltzmann 00:27:52.070 --> 00:27:53.750 align:middle line:90% distribution. 00:27:53.750 --> 00:27:56.930 align:middle line:90% There's another implication. 00:27:56.930 --> 00:27:59.660 align:middle line:84% The last thing I want to cover with only a couple of slides 00:27:59.660 --> 00:28:02.900 align:middle line:84% to go is the concept of ensembles, which 00:28:02.900 --> 00:28:06.920 align:middle line:90% is really just 3030 readiness. 00:28:06.920 --> 00:28:09.897 align:middle line:84% We're doing this just to show you that some concepts, 00:28:09.897 --> 00:28:11.730 align:middle line:84% which you're not going to use in this class, 00:28:11.730 --> 00:28:17.930 align:middle line:84% but you will use in kinetics and microstructure in the fall. 00:28:17.930 --> 00:28:21.080 align:middle line:84% So before we move on to discuss ensembles, 00:28:21.080 --> 00:28:23.240 align:middle line:84% I want to pause and take questions on the Boltzmann 00:28:23.240 --> 00:28:25.340 align:middle line:84% distribution, or Maxwell-Boltzmann 00:28:25.340 --> 00:28:29.030 align:middle line:84% or anything else, Arrhenius or anything else related. 00:28:29.030 --> 00:28:33.350 align:middle line:84% Who is seeing the Boltzmann distribution or this-- 00:28:33.350 --> 00:28:37.680 align:middle line:84% who's seen the Arrhenius rate law? 00:28:37.680 --> 00:28:40.260 align:middle line:84% There's not a lot of participation on Zoom, 00:28:40.260 --> 00:28:42.360 align:middle line:90% so I understand that. 00:28:42.360 --> 00:28:45.378 align:middle line:84% I'm guessing that a large number of people raised their hands. 00:28:45.378 --> 00:28:47.670 align:middle line:84% And then if I ask you about the Boltzmann distribution, 00:28:47.670 --> 00:28:48.970 align:middle line:90% there'd be a small number. 00:28:48.970 --> 00:28:50.970 align:middle line:84% And if I ask you about the Maxwell-Boltzmann distribution, 00:28:50.970 --> 00:28:52.720 align:middle line:84% they'd probably be an even smaller number. 00:28:52.720 --> 00:28:55.260 align:middle line:90% 00:28:55.260 --> 00:28:58.440 align:middle line:84% But I bet that everyone's seen something of this form 00:28:58.440 --> 00:28:59.703 align:middle line:90% somewhere before. 00:28:59.703 --> 00:29:01.620 align:middle line:84% So I hope it's a little bit satisfying for you 00:29:01.620 --> 00:29:05.220 align:middle line:84% to see where it comes from and to understand the principles 00:29:05.220 --> 00:29:07.935 align:middle line:90% behind it, maximum entropy. 00:29:07.935 --> 00:29:10.590 align:middle line:90% 00:29:10.590 --> 00:29:12.930 align:middle line:90% So let's talk about ensembles. 00:29:12.930 --> 00:29:20.290 align:middle line:90% 00:29:20.290 --> 00:29:23.230 align:middle line:90% Ensembles are-- what? 00:29:23.230 --> 00:29:25.690 align:middle line:84% I looked at the different definition of the Cambridge 00:29:25.690 --> 00:29:37.620 align:middle line:84% dictionary, says-- a group of things or people 00:29:37.620 --> 00:29:45.620 align:middle line:84% acting together or taken as a whole. 00:29:45.620 --> 00:29:50.900 align:middle line:90% 00:29:50.900 --> 00:29:56.020 align:middle line:84% So this is a concept, and we're going 00:29:56.020 --> 00:29:59.527 align:middle line:84% to use it a lot in statistical thermodynamics, if not 00:29:59.527 --> 00:30:00.610 align:middle line:90% necessarily in this class. 00:30:00.610 --> 00:30:08.218 align:middle line:84% But like, I said this is 3030 readiness here. 00:30:08.218 --> 00:30:09.760 align:middle line:84% If I didn't tell you about this, I've 00:30:09.760 --> 00:30:12.140 align:middle line:84% got in trouble with Professor Hu. 00:30:12.140 --> 00:30:12.640 align:middle line:90% All right. 00:30:12.640 --> 00:30:14.620 align:middle line:84% So we're going to start with the microcanonical ensemble. 00:30:14.620 --> 00:30:15.995 align:middle line:84% And this is something that you've 00:30:15.995 --> 00:30:18.064 align:middle line:90% seen before, microcanonical-- 00:30:18.064 --> 00:30:21.020 align:middle line:90% 00:30:21.020 --> 00:30:22.925 align:middle line:84% something called the microcanonical ensemble. 00:30:22.925 --> 00:30:27.910 align:middle line:90% 00:30:27.910 --> 00:30:39.020 align:middle line:84% This is the set of all possible microstates. 00:30:39.020 --> 00:30:41.930 align:middle line:84% So this type of stuff that we've been doing, 00:30:41.930 --> 00:30:51.680 align:middle line:84% set of all possible microstates of a system 00:30:51.680 --> 00:31:02.430 align:middle line:84% with fixed energy, volume, and particle number. 00:31:02.430 --> 00:31:04.890 align:middle line:84% So a system with fixed energy, volume, and particle number, 00:31:04.890 --> 00:31:06.960 align:middle line:84% that is that's an isolated system. 00:31:06.960 --> 00:31:11.640 align:middle line:90% 00:31:11.640 --> 00:31:13.170 align:middle line:84% For an isolated system equilibrium 00:31:13.170 --> 00:31:15.000 align:middle line:90% is state of max entropy. 00:31:15.000 --> 00:31:17.050 align:middle line:84% That's what we were just analyzing. 00:31:17.050 --> 00:31:17.550 align:middle line:90% All right. 00:31:17.550 --> 00:31:19.020 align:middle line:84% So this thing, which we've been doing 00:31:19.020 --> 00:31:20.562 align:middle line:84% for the last 2 and 1/2 lectures, this 00:31:20.562 --> 00:31:22.500 align:middle line:84% is called the microcanonical ensemble. 00:31:22.500 --> 00:31:23.688 align:middle line:90% Just gave a name to it. 00:31:23.688 --> 00:31:25.980 align:middle line:84% And so the partition function that we've been analyzing 00:31:25.980 --> 00:31:33.040 align:middle line:84% is actually better known as the microcanonical partition 00:31:33.040 --> 00:31:42.560 align:middle line:84% function, normally written as Q, as we've been doing. 00:31:42.560 --> 00:31:49.550 align:middle line:84% And it is a sum over single-particle energy states 00:31:49.550 --> 00:32:03.735 align:middle line:84% sum over all possible single-particle state. 00:32:03.735 --> 00:32:13.430 align:middle line:90% 00:32:13.430 --> 00:32:17.990 align:middle line:84% And we know that we're at equilibrium 00:32:17.990 --> 00:32:27.110 align:middle line:90% with S maximize for what? 00:32:27.110 --> 00:32:28.490 align:middle line:90% For the Boltzmann distribution. 00:32:28.490 --> 00:32:35.060 align:middle line:90% 00:32:35.060 --> 00:32:42.050 align:middle line:84% So the probability of finding particle in state I equals 00:32:42.050 --> 00:32:45.020 align:middle line:84% caused number of particles in state I at that distribution 00:32:45.020 --> 00:32:49.130 align:middle line:84% over the total number of particles equals e to the minus 00:32:49.130 --> 00:32:54.690 align:middle line:84% eI-- you are getting sick of seeing me write this by now all 00:32:54.690 --> 00:32:55.570 align:middle line:90% right-- 00:32:55.570 --> 00:33:09.590 align:middle line:84% prob of finding a single particle in 00:33:09.590 --> 00:33:19.465 align:middle line:90% state I with energy hereby. 00:33:19.465 --> 00:33:23.330 align:middle line:90% 00:33:23.330 --> 00:33:25.817 align:middle line:84% So this is a repeat of everything we just did. 00:33:25.817 --> 00:33:26.900 align:middle line:90% I just gave it a new name. 00:33:26.900 --> 00:33:28.580 align:middle line:90% I called it microcanonical. 00:33:28.580 --> 00:33:32.120 align:middle line:84% So again, that name is kind of highfalutin, 00:33:32.120 --> 00:33:35.350 align:middle line:90% but there you have it. 00:33:35.350 --> 00:33:41.510 align:middle line:90% But we can generalize. 00:33:41.510 --> 00:33:43.700 align:middle line:84% So now I'm going to tell you about a different type 00:33:43.700 --> 00:33:44.630 align:middle line:90% of ensemble. 00:33:44.630 --> 00:33:46.675 align:middle line:84% So we've been analyzing microcanonical. 00:33:46.675 --> 00:33:48.050 align:middle line:84% Now, I'm simply going to tell you 00:33:48.050 --> 00:33:51.860 align:middle line:84% about the existence of a canonical ensemble. 00:33:51.860 --> 00:33:54.500 align:middle line:90% 00:33:54.500 --> 00:33:58.470 align:middle line:84% And please, don't ask me where these names come from. 00:33:58.470 --> 00:34:17.500 align:middle line:84% A canonical ensemble is the set of all possible microstates 00:34:17.500 --> 00:34:38.517 align:middle line:84% of a system with fixed volume and n but not energy. 00:34:38.517 --> 00:34:40.684 align:middle line:84% So now we're allowing the system to exchange energy. 00:34:40.684 --> 00:34:45.219 align:middle line:90% 00:34:45.219 --> 00:34:47.160 align:middle line:90% The system is closed. 00:34:47.160 --> 00:34:50.260 align:middle line:90% 00:34:50.260 --> 00:34:54.600 align:middle line:90% It's rigid and closed. 00:34:54.600 --> 00:34:56.880 align:middle line:84% So volume and mass are conserved. 00:34:56.880 --> 00:35:12.440 align:middle line:84% System is rigid and closed but can exchange energy and form 00:35:12.440 --> 00:35:16.630 align:middle line:90% of heat with surroundings. 00:35:16.630 --> 00:35:24.090 align:middle line:90% 00:35:24.090 --> 00:35:28.910 align:middle line:84% And the system energy, which we're now 00:35:28.910 --> 00:35:35.450 align:middle line:84% going to give a label to, u sub nu, 00:35:35.450 --> 00:35:37.790 align:middle line:90% the system energy can fluctuate. 00:35:37.790 --> 00:35:43.410 align:middle line:90% 00:35:43.410 --> 00:35:46.520 align:middle line:84% So I want to draw for you a representation of the micro 00:35:46.520 --> 00:35:47.960 align:middle line:90% of the canonical ensemble. 00:35:47.960 --> 00:35:49.890 align:middle line:90% So we're going to imagine-- 00:35:49.890 --> 00:35:53.780 align:middle line:90% I grab a fine tip marker-- 00:35:53.780 --> 00:36:03.930 align:middle line:84% we're going to imagine system labeled I with energy u sub I, 00:36:03.930 --> 00:36:07.800 align:middle line:84% but this thing is in thermal contact 00:36:07.800 --> 00:36:11.105 align:middle line:90% with a lot of other systems. 00:36:11.105 --> 00:36:17.710 align:middle line:90% 00:36:17.710 --> 00:36:25.880 align:middle line:84% So we're going to imagine a whole population, or, if you 00:36:25.880 --> 00:36:35.480 align:middle line:84% like, an ensemble of systems all with fixed volume 00:36:35.480 --> 00:36:41.795 align:middle line:84% and fixed particle number but with diathermal walls. 00:36:41.795 --> 00:36:50.793 align:middle line:90% 00:36:50.793 --> 00:36:52.210 align:middle line:84% So if heat is like sound, you have 00:36:52.210 --> 00:36:54.760 align:middle line:84% to imagine this is an apartment building that's really noisy. 00:36:54.760 --> 00:36:55.630 align:middle line:90% Here are all your neighbors. 00:36:55.630 --> 00:36:57.970 align:middle line:84% So the volume is fixed and the particle number is fixed, 00:36:57.970 --> 00:37:01.270 align:middle line:84% but you're exchanging energy through the walls. 00:37:01.270 --> 00:37:03.610 align:middle line:90% But this ensemble as a whole-- 00:37:03.610 --> 00:37:06.890 align:middle line:90% 00:37:06.890 --> 00:37:08.170 align:middle line:90% so I've drawn an ensemble. 00:37:08.170 --> 00:37:11.920 align:middle line:84% I've drawn a set or a group considered as a whole. 00:37:11.920 --> 00:37:14.830 align:middle line:84% I've drawn a bunch of subsystems now 00:37:14.830 --> 00:37:18.490 align:middle line:84% that can exchange energy with each other. 00:37:18.490 --> 00:37:19.685 align:middle line:90% Each have the same volume. 00:37:19.685 --> 00:37:21.310 align:middle line:84% Well, that's how I've drawn it, but you 00:37:21.310 --> 00:37:23.170 align:middle line:84% can imagine they each have the same volume, 00:37:23.170 --> 00:37:25.210 align:middle line:84% each have the same particle number, 00:37:25.210 --> 00:37:34.980 align:middle line:84% but the ensemble as a whole is isolated. 00:37:34.980 --> 00:37:38.852 align:middle line:90% 00:37:38.852 --> 00:37:40.310 align:middle line:84% So all of these ensembles, you have 00:37:40.310 --> 00:37:42.685 align:middle line:84% to imagine this being a very, very large number here, not 00:37:42.685 --> 00:37:50.540 align:middle line:84% just 4 by 6 or whatever I've drawn, is isolated. 00:37:50.540 --> 00:37:53.420 align:middle line:84% And so this is an ensemble, and this 00:37:53.420 --> 00:37:55.730 align:middle line:84% is an ensemble that has its own partition function. 00:37:55.730 --> 00:38:00.540 align:middle line:84% It's known as the canonical partition function. 00:38:00.540 --> 00:38:06.230 align:middle line:90% 00:38:06.230 --> 00:38:10.040 align:middle line:84% And that's typically written a z, and this is now sum-- 00:38:10.040 --> 00:38:13.590 align:middle line:90% 00:38:13.590 --> 00:38:15.330 align:middle line:90% as before-- sum over states. 00:38:15.330 --> 00:38:17.760 align:middle line:84% But we're not summing over single-particle states. 00:38:17.760 --> 00:38:26.000 align:middle line:84% We're summing over system energy states, 00:38:26.000 --> 00:38:39.845 align:middle line:84% sum over all possible states of the system. 00:38:39.845 --> 00:38:45.290 align:middle line:90% 00:38:45.290 --> 00:38:46.750 align:middle line:84% So it's distinct from before when 00:38:46.750 --> 00:38:49.090 align:middle line:84% we were coming over single-particle states 00:38:49.090 --> 00:39:00.010 align:middle line:90% and equilibrium Hold on. 00:39:00.010 --> 00:39:02.560 align:middle line:84% When we had equilibrium in an isolated system, 00:39:02.560 --> 00:39:04.940 align:middle line:90% it was max entropy. 00:39:04.940 --> 00:39:06.650 align:middle line:90% Now, think back two months ago. 00:39:06.650 --> 00:39:10.280 align:middle line:84% When we had equilibrium for fixed temperature and fixed 00:39:10.280 --> 00:39:14.750 align:middle line:84% pressure, that was minimum Gibbs, 00:39:14.750 --> 00:39:16.850 align:middle line:84% but there was another case, which we 00:39:16.850 --> 00:39:18.200 align:middle line:90% haven't spent a lot of time on. 00:39:18.200 --> 00:39:22.550 align:middle line:84% It was equilibrium at fixed volume and fixed temperature. 00:39:22.550 --> 00:39:24.500 align:middle line:84% Do you remember what thermodynamic potential 00:39:24.500 --> 00:39:27.890 align:middle line:84% we use when we have a system at fixed volume and fixed 00:39:27.890 --> 00:39:31.030 align:middle line:90% temperature? 00:39:31.030 --> 00:39:32.113 align:middle line:90% STUDENT: Is it Helmholtz? 00:39:32.113 --> 00:39:33.280 align:middle line:90% RAFAEL JARAMILLO: Helmholtz. 00:39:33.280 --> 00:39:37.450 align:middle line:84% Equilibrium at minimum F, again, we 00:39:37.450 --> 00:39:39.820 align:middle line:84% haven't used that, this course basically, 00:39:39.820 --> 00:39:49.740 align:middle line:84% is minimized for the distribution 00:39:49.740 --> 00:39:55.140 align:middle line:84% probability of finding system and state nu equals e 00:39:55.140 --> 00:40:05.090 align:middle line:84% to the minus u nu over kT over Z. 00:40:05.090 --> 00:40:08.930 align:middle line:84% So has the same functional form as the bottom distribution 00:40:08.930 --> 00:40:12.590 align:middle line:84% except, instead of single-particle energies is 00:40:12.590 --> 00:40:15.290 align:middle line:84% energy of the system as a whole, and instead 00:40:15.290 --> 00:40:20.550 align:middle line:84% of maximizing entropy, it minimizes how much free energy. 00:40:20.550 --> 00:40:22.290 align:middle line:84% So there's something very similar 00:40:22.290 --> 00:40:24.260 align:middle line:84% That's happening here, similar to when 00:40:24.260 --> 00:40:27.830 align:middle line:84% we started with entropy, and then we move to Helmholtz 00:40:27.830 --> 00:40:30.170 align:middle line:84% energy, and then we moved to Gibbs energy 00:40:30.170 --> 00:40:32.480 align:middle line:84% as we change the boundary conditions of the system. 00:40:32.480 --> 00:40:34.355 align:middle line:84% And I think you'll remember in problem set 2, 00:40:34.355 --> 00:40:36.755 align:middle line:90% we really delve into this. 00:40:36.755 --> 00:40:38.130 align:middle line:84% So there's something very similar 00:40:38.130 --> 00:40:40.560 align:middle line:84% that happens in statistical thermodynamics, where 00:40:40.560 --> 00:40:42.420 align:middle line:84% you start with the microcanonical ensemble 00:40:42.420 --> 00:40:44.058 align:middle line:84% and its partition function, and then 00:40:44.058 --> 00:40:46.600 align:middle line:84% you go to the canonical ensemble and it's partition function, 00:40:46.600 --> 00:40:47.310 align:middle line:90% and there's more. 00:40:47.310 --> 00:40:48.393 align:middle line:90% There are other ensembles. 00:40:48.393 --> 00:40:50.730 align:middle line:84% The next one is called the grand canonical ensemble, 00:40:50.730 --> 00:40:54.000 align:middle line:84% and that corresponds to situations with fixed pressure 00:40:54.000 --> 00:40:54.820 align:middle line:90% and temperature. 00:40:54.820 --> 00:40:58.110 align:middle line:84% We're not going to bother to introduce that. 00:40:58.110 --> 00:41:05.780 align:middle line:84% So this is the Boltzmann distribution, 00:41:05.780 --> 00:41:09.830 align:middle line:84% but is for the canonical ensemble now. 00:41:09.830 --> 00:41:14.100 align:middle line:90% 00:41:14.100 --> 00:41:16.965 align:middle line:84% STUDENT: So in this case, what was it that you held constant? 00:41:16.965 --> 00:41:19.743 align:middle line:90% 00:41:19.743 --> 00:41:21.410 align:middle line:84% RAFAEL JARAMILLO: So this is equilibrium 00:41:21.410 --> 00:41:25.030 align:middle line:90% at fixed temperature. 00:41:25.030 --> 00:41:29.990 align:middle line:84% When you have an ensemble of systems 00:41:29.990 --> 00:41:35.000 align:middle line:84% that can't exchange volume and they can't exchange particle 00:41:35.000 --> 00:41:38.360 align:middle line:84% number, but they can exchange heat energy, 00:41:38.360 --> 00:41:44.440 align:middle line:84% they reach equilibrium when the temperature becomes constant. 00:41:44.440 --> 00:41:46.178 align:middle line:90% That's thermal equilibrium. 00:41:46.178 --> 00:41:47.470 align:middle line:90% So that's what we've done here. 00:41:47.470 --> 00:41:50.470 align:middle line:84% We've set up a system, a collection and ensemble 00:41:50.470 --> 00:41:53.980 align:middle line:84% of subsystems, that reach thermal equilibrium 00:41:53.980 --> 00:41:58.630 align:middle line:84% with each other because they can exchange heat energy. 00:41:58.630 --> 00:42:00.160 align:middle line:90% It all works out. 00:42:00.160 --> 00:42:04.810 align:middle line:84% You end up getting minimum F for this distribution. 00:42:04.810 --> 00:42:20.700 align:middle line:84% This is the probability of finding system in state nu 00:42:20.700 --> 00:42:29.290 align:middle line:90% with energy u of nu. 00:42:29.290 --> 00:42:32.920 align:middle line:90% That's what that probability is. 00:42:32.920 --> 00:42:35.650 align:middle line:84% So I want to leave you with some final words on partition 00:42:35.650 --> 00:42:38.350 align:middle line:90% functions. 00:42:38.350 --> 00:42:48.020 align:middle line:84% We've seen some partition functions, Q and Z. 00:42:48.020 --> 00:42:51.198 align:middle line:84% And there are others, which we won't cover in this class. 00:42:51.198 --> 00:42:52.490 align:middle line:90% We haven't really covered this. 00:42:52.490 --> 00:42:53.920 align:middle line:90% This won't be on the exam. 00:42:53.920 --> 00:42:56.420 align:middle line:84% This is just commenting on what comes next. 00:42:56.420 --> 00:42:59.660 align:middle line:84% Again, this is in service to later classes 00:42:59.660 --> 00:43:00.570 align:middle line:90% in our curriculum. 00:43:00.570 --> 00:43:06.710 align:middle line:84% So the partition functions, they describe 00:43:06.710 --> 00:43:13.770 align:middle line:90% equilibrium and fluctuations. 00:43:13.770 --> 00:43:17.630 align:middle line:90% 00:43:17.630 --> 00:43:19.550 align:middle line:84% The partition functions really describe 00:43:19.550 --> 00:43:21.785 align:middle line:84% the entire thermodynamic properties of a system. 00:43:21.785 --> 00:43:26.530 align:middle line:90% 00:43:26.530 --> 00:43:31.560 align:middle line:84% So for instance, they are the basis 00:43:31.560 --> 00:43:49.275 align:middle line:84% for calculating thermodynamic properties and expectation 00:43:49.275 --> 00:43:49.775 align:middle line:90% values. 00:43:49.775 --> 00:43:55.310 align:middle line:90% 00:43:55.310 --> 00:43:56.790 align:middle line:90% That is, most likely quantity. 00:43:56.790 --> 00:44:03.180 align:middle line:84% So for example, the mean energy of a system 00:44:03.180 --> 00:44:09.350 align:middle line:84% can be shown to be this derivative, d log 00:44:09.350 --> 00:44:15.032 align:middle line:90% z, d 1 over kT. 00:44:15.032 --> 00:44:17.480 align:middle line:84% That's a little bit formal, can't necessarily 00:44:17.480 --> 00:44:19.322 align:middle line:84% calculate anything yet, but in principle, 00:44:19.322 --> 00:44:20.780 align:middle line:84% if you know the partition function, 00:44:20.780 --> 00:44:24.000 align:middle line:84% you can calculate the mean energy. 00:44:24.000 --> 00:44:33.640 align:middle line:84% heat capacity of a system is 1 over kT squared d squared log 00:44:33.640 --> 00:44:38.890 align:middle line:90% z d 1 over kT squared. 00:44:38.890 --> 00:44:41.770 align:middle line:90% 00:44:41.770 --> 00:44:46.680 align:middle line:84% The entropy of a system can be shown 00:44:46.680 --> 00:44:56.150 align:middle line:84% to be mean energy divided by temperature plus kb log z. 00:44:56.150 --> 00:45:04.320 align:middle line:84% The Helmholtry energy of a system is minus KbT log z. 00:45:04.320 --> 00:45:06.900 align:middle line:84% And it's not just equilibrium properties, 00:45:06.900 --> 00:45:09.580 align:middle line:90% it's properties of fluctuations. 00:45:09.580 --> 00:45:13.590 align:middle line:84% So this gets really fun in a later class 00:45:13.590 --> 00:45:15.580 align:middle line:90% and has a lot of implications. 00:45:15.580 --> 00:45:20.310 align:middle line:84% So for example, the root mean squared energy fluctuation 00:45:20.310 --> 00:45:30.790 align:middle line:84% of a system is another quantity that can be calculated 00:45:30.790 --> 00:45:33.140 align:middle line:90% from partition functions. 00:45:33.140 --> 00:45:37.150 align:middle line:84% So I'm just showing you things, but if you have the partition 00:45:37.150 --> 00:45:41.560 align:middle line:84% function for a system, you can calculate 00:45:41.560 --> 00:45:44.390 align:middle line:84% the entire thermodynamic properties of that system. 00:45:44.390 --> 00:45:46.510 align:middle line:84% So another example is in the textbook. 00:45:46.510 --> 00:45:50.028 align:middle line:84% If you look into Hoff, they calculate the partition 00:45:50.028 --> 00:45:51.820 align:middle line:84% function for a couple of different systems, 00:45:51.820 --> 00:45:57.660 align:middle line:84% a two-level system and then a system of a monatomic gas. 00:45:57.660 --> 00:46:00.220 align:middle line:84% And then by applying formula such as these, 00:46:00.220 --> 00:46:02.640 align:middle line:90% they derive the ideal gas law. 00:46:02.640 --> 00:46:05.470 align:middle line:90% 00:46:05.470 --> 00:46:08.340 align:middle line:84% So it's the ideal gas law from first principles. 00:46:08.340 --> 00:46:09.540 align:middle line:90% It's no longer empirical. 00:46:09.540 --> 00:46:12.530 align:middle line:90% 00:46:12.530 --> 00:46:16.280 align:middle line:84% So I'm going to leave you with some thoughts. 00:46:16.280 --> 00:46:20.960 align:middle line:84% At this point, often students say, ah, 00:46:20.960 --> 00:46:22.620 align:middle line:90% this is the real stuff. 00:46:22.620 --> 00:46:27.420 align:middle line:84% This is somehow the most fundamental knowledge 00:46:27.420 --> 00:46:30.640 align:middle line:84% of thermodynamics, is a partition functions. 00:46:30.640 --> 00:46:33.390 align:middle line:84% So why did we mess around with all 00:46:33.390 --> 00:46:34.950 align:middle line:90% that classical thermodynamics? 00:46:34.950 --> 00:46:38.100 align:middle line:84% If the partition functions are the keys to the kingdom, 00:46:38.100 --> 00:46:41.300 align:middle line:90% why not just start there? 00:46:41.300 --> 00:46:45.130 align:middle line:84% And the answer is that, although this approach is 00:46:45.130 --> 00:46:48.280 align:middle line:84% beautiful and elegant and very powerful in instances, 00:46:48.280 --> 00:46:52.070 align:middle line:84% it also is extremely limited because we don't know 00:46:52.070 --> 00:46:56.050 align:middle line:84% and we can't calculate the partition functions for all 00:46:56.050 --> 00:47:00.180 align:middle line:90% but the most toy problems. 00:47:00.180 --> 00:47:03.680 align:middle line:90% And you can see that. 00:47:03.680 --> 00:47:06.480 align:middle line:84% If we take thermodynamics in a physics department, 00:47:06.480 --> 00:47:07.830 align:middle line:90% you start from this approach. 00:47:07.830 --> 00:47:10.440 align:middle line:84% You start from the statistical thermodynamics approach 00:47:10.440 --> 00:47:14.490 align:middle line:84% because physicists love to write problems, 00:47:14.490 --> 00:47:17.370 align:middle line:84% things as intellectually rich but may or may not 00:47:17.370 --> 00:47:19.530 align:middle line:90% apply to the real world. 00:47:19.530 --> 00:47:21.750 align:middle line:90% We're a little bit more applied. 00:47:21.750 --> 00:47:23.460 align:middle line:90% So I'll give you an example. 00:47:23.460 --> 00:47:27.900 align:middle line:84% There's a toy model of magnetism known as the Ising model. 00:47:27.900 --> 00:47:29.520 align:middle line:84% And the Ising model says that you 00:47:29.520 --> 00:47:35.770 align:middle line:84% can have spins on a lattice, or magnetic moments, 00:47:35.770 --> 00:47:39.750 align:middle line:84% I should say, on a lattice, and they can face up or down. 00:47:39.750 --> 00:47:40.410 align:middle line:90% That's it. 00:47:40.410 --> 00:47:42.490 align:middle line:84% So you have a one-dimensional chain, 00:47:42.490 --> 00:47:44.580 align:middle line:84% and each spin can face up or down. 00:47:44.580 --> 00:47:47.540 align:middle line:84% And the energy of a system depends on the orientation 00:47:47.540 --> 00:47:48.540 align:middle line:90% of the nearest neighbor. 00:47:48.540 --> 00:47:51.970 align:middle line:84% It's a very simple model of a magnet. 00:47:51.970 --> 00:47:54.690 align:middle line:84% And so if you take a course on magnetism, 00:47:54.690 --> 00:47:57.370 align:middle line:84% or you take a thermodynamics course in physics department, 00:47:57.370 --> 00:47:59.370 align:middle line:84% you'll probably calculate the partition function 00:47:59.370 --> 00:48:01.398 align:middle line:90% for 1D Ising chain. 00:48:01.398 --> 00:48:02.190 align:middle line:90% It's not that hard. 00:48:02.190 --> 00:48:02.898 align:middle line:90% It's kind of fun. 00:48:02.898 --> 00:48:07.320 align:middle line:90% 00:48:07.320 --> 00:48:12.042 align:middle line:84% However, if we simply make it a two-dimensional Ising system-- 00:48:12.042 --> 00:48:14.250 align:middle line:84% and I'm drawing the direction of the arrows at random 00:48:14.250 --> 00:48:17.260 align:middle line:90% here, just showing you. 00:48:17.260 --> 00:48:17.790 align:middle line:90% All right. 00:48:17.790 --> 00:48:20.070 align:middle line:84% So we've just made this from one dimensional, which 00:48:20.070 --> 00:48:21.960 align:middle line:84% is definitely an imaginary thing, 00:48:21.960 --> 00:48:24.980 align:middle line:84% to a two dimensional, which is still imaginary, 00:48:24.980 --> 00:48:25.980 align:middle line:90% but we're getting there. 00:48:25.980 --> 00:48:28.080 align:middle line:84% We're at least getting close to real space. 00:48:28.080 --> 00:48:33.060 align:middle line:84% Calculating the partition function for this model 00:48:33.060 --> 00:48:36.390 align:middle line:84% is one of the grand intellectual achievements of 20th century 00:48:36.390 --> 00:48:39.380 align:middle line:84% physics and, it won Lars Unger a Nobel Prize. 00:48:39.380 --> 00:48:42.830 align:middle line:84% It's horrendously complicated mathematically. 00:48:42.830 --> 00:48:46.340 align:middle line:84% I've never been able to follow even the first page of that. 00:48:46.340 --> 00:48:47.540 align:middle line:90% It's not in any textbooks. 00:48:47.540 --> 00:48:48.155 align:middle line:90% It's too long. 00:48:48.155 --> 00:48:51.520 align:middle line:90% 00:48:51.520 --> 00:48:54.450 align:middle line:84% So let's say I want to model a real magnet in three 00:48:54.450 --> 00:48:55.470 align:middle line:90% dimensions. 00:48:55.470 --> 00:48:56.250 align:middle line:90% Forget it. 00:48:56.250 --> 00:48:59.645 align:middle line:84% Unless you're Nobel Prize quality, that's an open problem 00:48:59.645 --> 00:49:00.270 align:middle line:90% you have at it. 00:49:00.270 --> 00:49:03.480 align:middle line:84% But basically, the point is that for real systems 00:49:03.480 --> 00:49:09.390 align:middle line:84% that we care about as engineers, this is a magical construct. 00:49:09.390 --> 00:49:11.460 align:middle line:90% In theory, this exists. 00:49:11.460 --> 00:49:13.830 align:middle line:84% In theory, it allows you to calculate all the properties 00:49:13.830 --> 00:49:15.450 align:middle line:84% of the system from first principles, 00:49:15.450 --> 00:49:20.090 align:middle line:84% but in practice, it just remains an intellectual exercise. 00:49:20.090 --> 00:49:21.560 align:middle line:90% In practice, we have to-- 00:49:21.560 --> 00:49:24.620 align:middle line:84% if you go with a much more powerful route, 00:49:24.620 --> 00:49:27.110 align:middle line:84% which is the route of classical thermodynamics 00:49:27.110 --> 00:49:30.800 align:middle line:84% with its postulates and observations and data 00:49:30.800 --> 00:49:32.680 align:middle line:90% and models. 00:49:32.680 --> 00:49:35.000 align:middle line:90%