WEBVTT 00:00:01.520 --> 00:00:03.890 The following content is provided under a Creative 00:00:03.890 --> 00:00:05.280 Commons license. 00:00:05.280 --> 00:00:07.520 Your support will help MIT OpenCourseWare 00:00:07.520 --> 00:00:11.610 continue to offer high-quality educational resources for free. 00:00:11.610 --> 00:00:14.150 To make a donation or to view additional materials 00:00:14.150 --> 00:00:18.080 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.080 --> 00:00:19.320 at ocw.mit.edu. 00:00:24.440 --> 00:00:27.500 WILLIAM GREEN, JR: All right, let's get going. 00:00:27.500 --> 00:00:32.490 So today, I'll say a few more words about PDEs, 00:00:32.490 --> 00:00:34.490 and then we'll leave that topic for a while. 00:00:34.490 --> 00:00:36.156 I think I'll actually come back and have 00:00:36.156 --> 00:00:39.530 a little lecture about that right before Thanksgiving, 00:00:39.530 --> 00:00:42.140 for those of you still around. 00:00:42.140 --> 00:00:44.110 And then we'll start talking about probability, 00:00:44.110 --> 00:00:47.000 and then that will lead into several lectures 00:00:47.000 --> 00:00:50.930 about models versus data, which is a very important topic 00:00:50.930 --> 00:00:54.080 for all of you who either plan to generate data 00:00:54.080 --> 00:00:57.290 or plan to generate models during your stay here, 00:00:57.290 --> 00:00:59.689 which probably is everybody. 00:00:59.689 --> 00:01:01.230 So we'll talk about that for a while. 00:01:03.830 --> 00:01:06.765 So PDEs. 00:01:06.765 --> 00:01:09.530 I guess the first comment is, homework 7, have any of you 00:01:09.530 --> 00:01:10.608 looked at this yet? 00:01:10.608 --> 00:01:11.922 AUDIENCE: [INTERPOSING VOICES] 00:01:11.922 --> 00:01:13.372 [LAUGHTER] 00:01:13.372 --> 00:01:14.871 WILLIAM GREEN, JR: Not a single one. 00:01:14.871 --> 00:01:16.344 AUDIENCE: [INAUDIBLE] 00:01:16.344 --> 00:01:19.781 [LAUGHTER] 00:01:22.250 --> 00:01:24.870 WILLIAM GREEN, JR: So homework 7 is the same problem 00:01:24.870 --> 00:01:29.280 that Kristin showed in class in the demo for COMSOL, 00:01:29.280 --> 00:01:30.880 and I want you to solve it both ways. 00:01:30.880 --> 00:01:33.270 So solve it with COMSOL and then solve 00:01:33.270 --> 00:01:35.770 it writing your own finite volume code. 00:01:35.770 --> 00:01:39.350 And I want to warn you. 00:01:39.350 --> 00:01:43.380 This problem has a characteristic length 00:01:43.380 --> 00:01:50.950 that's way smaller than the dimensions of the problem. 00:01:50.950 --> 00:01:54.270 And so in principle, you might need 00:01:54.270 --> 00:01:57.510 to use an incredibly fine mesh to resolve 00:01:57.510 --> 00:01:59.540 the gradients in the problem. 00:01:59.540 --> 00:02:03.600 So just remember, in case you haven't looked at it, 00:02:03.600 --> 00:02:05.670 the problem is we have a drug patch. 00:02:05.670 --> 00:02:07.770 It has some concentration of the drug. 00:02:07.770 --> 00:02:11.960 The drug is diffusing slowly out of the patch into the flow, 00:02:11.960 --> 00:02:15.720 and we have some flow here like that. 00:02:15.720 --> 00:02:18.342 And the characteristic length is at the-- 00:02:18.342 --> 00:02:20.950 this is a velocity boundary layer. 00:02:20.950 --> 00:02:23.670 So the velocity in the x-direction 00:02:23.670 --> 00:02:33.520 is equal to y times something, dv dy, I guess. 00:02:33.520 --> 00:02:37.590 This is some number, and so this has units of-- 00:02:40.180 --> 00:02:45.780 what is that-- per second, so like a strain rate, OK? 00:02:45.780 --> 00:02:51.210 And the diffusion here is controlled by diffusivity D, 00:02:51.210 --> 00:02:55.690 and that has units of, say, centimeter squared per second. 00:02:55.690 --> 00:03:04.710 And so the D over dvx dy, this thing 00:03:04.710 --> 00:03:10.350 gives a characteristic length squared, 00:03:10.350 --> 00:03:13.840 which is sort of the natural length scale of this problem. 00:03:13.840 --> 00:03:17.182 And the problem is that for the drug molecule, 00:03:17.182 --> 00:03:18.015 it's a big molecule. 00:03:18.015 --> 00:03:20.390 It has a very small diffusivity. 00:03:20.390 --> 00:03:23.740 And so therefore, this is a really tiny ratio, 00:03:23.740 --> 00:03:25.750 and so the L is very small. 00:03:25.750 --> 00:03:29.400 And similarly, if you look at it from the point 00:03:29.400 --> 00:03:31.800 of view in the x-direction, the Peclet number 00:03:31.800 --> 00:03:33.417 is really gigantic. 00:03:33.417 --> 00:03:36.000 And so both of those will tell you that you have to watch out, 00:03:36.000 --> 00:03:38.125 there might be very sharp gradients in the problem. 00:03:38.125 --> 00:03:41.400 And if you just think of it physically, right over here 00:03:41.400 --> 00:03:44.189 we think the concentration's 0, somewhere over here. 00:03:44.189 --> 00:03:46.230 And all of a sudden right here, the concentration 00:03:46.230 --> 00:03:48.600 is going to be close to the concentration in the patch. 00:03:48.600 --> 00:03:49.980 So there's almost a discontinuity 00:03:49.980 --> 00:03:51.730 in the concentration. 00:03:51.730 --> 00:03:55.032 So there's a really sharp gradient on the upstream edge. 00:03:55.032 --> 00:03:57.240 And then something funky is going to happen down here 00:03:57.240 --> 00:03:58.070 at the end of the patch, too. 00:03:58.070 --> 00:04:00.660 It won't be quite as abrupt, but could be pretty strange. 00:04:03.210 --> 00:04:04.370 All right, so you got it? 00:04:04.370 --> 00:04:04.870 OK. 00:04:07.546 --> 00:04:09.420 And in the problem, we want you to figure out 00:04:09.420 --> 00:04:12.590 the drug diffusing all the way over to here somewhere, 00:04:12.590 --> 00:04:14.010 way far over there. 00:04:14.010 --> 00:04:17.670 And so you may need quite a few mesh points in the y-direction 00:04:17.670 --> 00:04:19.330 as well. 00:04:19.330 --> 00:04:19.829 All right. 00:04:19.829 --> 00:04:24.690 And this kind of problem, this is a very simple case, right? 00:04:24.690 --> 00:04:26.970 There's no reactions. 00:04:26.970 --> 00:04:28.725 The velocity's just in one direction, 00:04:28.725 --> 00:04:30.414 and this is not a very hard case. 00:04:30.414 --> 00:04:31.830 But you'll see it's actually still 00:04:31.830 --> 00:04:33.910 pretty tricky to get the right solution. 00:04:33.910 --> 00:04:36.660 So don't just believe what the code tells you. 00:04:36.660 --> 00:04:38.580 Just run COMSOL and just-- 00:04:38.580 --> 00:04:40.860 don't believe it's showing the truth. 00:04:40.860 --> 00:04:43.770 And don't believe you just write down some finite volumes 00:04:43.770 --> 00:04:45.045 that you'll get the truth. 00:04:45.045 --> 00:04:47.170 So mess around with it and try to convince yourself 00:04:47.170 --> 00:04:48.090 it's really converged, and you really 00:04:48.090 --> 00:04:49.423 have the real physical solution. 00:04:51.532 --> 00:04:52.990 Because we expect a sharp gradient, 00:04:52.990 --> 00:04:54.660 say, in the upstream edge, you might 00:04:54.660 --> 00:04:58.590 want to play with using a finer mesh in the x-direction 00:04:58.590 --> 00:05:02.350 here than you would down here, because down here presumably 00:05:02.350 --> 00:05:04.560 the gradients in x-direction are much smaller. 00:05:04.560 --> 00:05:08.490 So you don't have to use square finite volumes. 00:05:08.490 --> 00:05:09.960 You could use rectangles. 00:05:09.960 --> 00:05:11.005 Yes? 00:05:11.005 --> 00:05:12.380 AUDIENCE: I don't understand what 00:05:12.380 --> 00:05:15.736 you have written [INAUDIBLE] as vx equal y dvx dy. 00:05:15.736 --> 00:05:16.736 WILLIAM GREEN, JR: Yeah. 00:05:16.736 --> 00:05:18.680 AUDIENCE: So what-- 00:05:18.680 --> 00:05:21.096 WILLIAM GREEN, JR: So that's because the velocity-- the vx 00:05:21.096 --> 00:05:27.490 is 0 at the wall, and the velocity increases with y. 00:05:27.490 --> 00:05:30.770 So vx increases the further you get from y. 00:05:30.770 --> 00:05:34.870 This is y equals 0, always going up that way. 00:05:34.870 --> 00:05:37.930 As you can increase the height, the flow gets faster. 00:05:37.930 --> 00:05:40.330 And that's typical, because you have a no-slip boundary 00:05:40.330 --> 00:05:43.330 condition at the wall, right? 00:05:43.330 --> 00:05:46.681 Is that OK, or did I misunderstand your question? 00:05:46.681 --> 00:05:49.680 Is that good? 00:05:49.680 --> 00:05:50.180 OK? 00:05:50.180 --> 00:05:51.950 AUDIENCE: So y is a constant. 00:05:51.950 --> 00:05:53.270 WILLIAM GREEN, JR: Yeah, in this particular problem 00:05:53.270 --> 00:05:55.720 this is just a number that we tell you in the problem. 00:05:55.720 --> 00:05:58.539 AUDIENCE: [INAUDIBLE] 00:05:58.539 --> 00:05:59.580 WILLIAM GREEN, JR: Right. 00:05:59.580 --> 00:06:01.246 If you had a flow in a cylindrical pipe, 00:06:01.246 --> 00:06:02.260 it wouldn't be-- 00:06:02.260 --> 00:06:03.140 wouldn't be a number. 00:06:03.140 --> 00:06:06.807 Be more complicated, yeah, all right? 00:06:06.807 --> 00:06:09.390 So this is-- I mean, again, this is like the simplest you got. 00:06:09.390 --> 00:06:11.099 It's only 2D. 00:06:11.099 --> 00:06:11.890 It's really simple. 00:06:11.890 --> 00:06:13.465 But you're going to see here that even here you 00:06:13.465 --> 00:06:14.631 could have a lot of trouble. 00:06:14.631 --> 00:06:16.360 And if you just don't think about it-- 00:06:16.360 --> 00:06:17.560 the computer will give you an answer. 00:06:17.560 --> 00:06:19.680 Whatever-- you put in some finite difference equations 00:06:19.680 --> 00:06:21.429 or finite element equations, finite volume 00:06:21.429 --> 00:06:23.542 equations, fsolve, whatever, it's 00:06:23.542 --> 00:06:25.250 going to solve and give you some numbers. 00:06:25.250 --> 00:06:26.350 It doesn't mean it has any relation 00:06:26.350 --> 00:06:27.350 to the physical reality. 00:06:27.350 --> 00:06:29.860 So this is a problem to really pay attention 00:06:29.860 --> 00:06:33.230 to whether you're really converged. 00:06:33.230 --> 00:06:35.142 All right. 00:06:35.142 --> 00:06:39.330 And so what I suggest you do in this problem is go 00:06:39.330 --> 00:06:42.415 through a sequence where you vary-- 00:06:42.415 --> 00:06:45.040 the real problem wants you to do this all the way out to-- this 00:06:45.040 --> 00:06:47.237 is 1 centimeter. 00:06:47.237 --> 00:06:49.320 But I suggest you instead solve a simpler problem, 00:06:49.320 --> 00:06:51.194 where you put the-- here's a simpler problem. 00:06:54.490 --> 00:07:02.330 Put your boundary right here, really close, and then 00:07:02.330 --> 00:07:03.470 solve that problem. 00:07:03.470 --> 00:07:07.650 You won't need as many mesh points across 00:07:07.650 --> 00:07:09.150 to get from here to here. 00:07:09.150 --> 00:07:10.140 And this should be-- 00:07:10.140 --> 00:07:12.420 this is-- suppose you choose this 00:07:12.420 --> 00:07:16.420 is c equal 0 boundary condition for this wall, 00:07:16.420 --> 00:07:19.840 then this should be an upper bound. 00:07:19.840 --> 00:07:22.540 Because if you put an absorbing layer here, 00:07:22.540 --> 00:07:25.930 it should drive the diffusion faster, right? 00:07:25.930 --> 00:07:30.356 So as you increase this distance-- 00:07:30.356 --> 00:07:32.260 let's call it little h-- 00:07:32.260 --> 00:07:34.697 as you increase little h, you should 00:07:34.697 --> 00:07:36.280 converge to the true solution that you 00:07:36.280 --> 00:07:39.252 want sort of from above. 00:07:39.252 --> 00:07:41.641 Does that makes sense? 00:07:41.641 --> 00:07:43.072 Yeah? 00:07:43.072 --> 00:07:45.230 Is this OK? 00:07:45.230 --> 00:07:46.730 All right. 00:07:46.730 --> 00:07:50.330 And similarly, you can vary the mesh, 00:07:50.330 --> 00:07:53.666 how big your boxes are, say, in delta x and delta y. 00:07:53.666 --> 00:07:55.040 And again, for each of those, you 00:07:55.040 --> 00:07:58.070 could think about how should things converge. 00:07:58.070 --> 00:07:59.690 And so look and see if it's actually 00:07:59.690 --> 00:08:02.516 converging the way you think it should be converging, right? 00:08:06.460 --> 00:08:08.589 Any more questions about this? 00:08:08.589 --> 00:08:10.630 OK, so this problem might-- this homework problem 00:08:10.630 --> 00:08:13.119 might look like a MATLAB coding problem. 00:08:13.119 --> 00:08:14.910 It has MATLAB coding, but it's not really-- 00:08:14.910 --> 00:08:16.237 that's not really what it is. 00:08:16.237 --> 00:08:17.070 It's really like a-- 00:08:17.070 --> 00:08:20.113 it's a conceptual problem about what you're doing. 00:08:20.113 --> 00:08:22.879 All right. 00:08:22.879 --> 00:08:26.020 All right, in this problem, I want you to use finite volumes. 00:08:26.020 --> 00:08:28.020 So let's talk about finite volumes for a minute. 00:08:28.020 --> 00:08:33.890 So finite volumes is to imagine that we have little control 00:08:33.890 --> 00:08:38.880 volumes, and each one of them has a little propeller in it, 00:08:38.880 --> 00:08:43.200 stirs it up, like little CSTRs. 00:08:43.200 --> 00:08:45.670 And so we do it just like you did back in your intro ChemE 00:08:45.670 --> 00:08:48.820 class and mass/energy balances a million years ago. 00:08:48.820 --> 00:08:51.640 You know that you have some flow coming in here, 00:08:51.640 --> 00:08:54.120 and maybe some flow going out there. 00:08:54.120 --> 00:08:56.895 And maybe you have some flux come in here 00:08:56.895 --> 00:08:59.370 and maybe something out there. 00:08:59.370 --> 00:09:02.400 And you can add up all the flows in and all the flows out, 00:09:02.400 --> 00:09:05.350 and then that-- 00:09:05.350 --> 00:09:08.750 the net of all these fluxes has got 00:09:08.750 --> 00:09:10.560 to be equal to the accumulation. 00:09:10.560 --> 00:09:12.480 And if we're doing a steady-state problem, 00:09:12.480 --> 00:09:15.600 then there's no time derivative, so the accumulation term 00:09:15.600 --> 00:09:16.662 should be 0. 00:09:16.662 --> 00:09:18.120 And you did a lot of these problems 00:09:18.120 --> 00:09:19.920 a long time ago, right? 00:09:19.920 --> 00:09:21.296 OK, so you just do it like that. 00:09:21.296 --> 00:09:23.920 The only problem is now we have a million of these little boxes 00:09:23.920 --> 00:09:25.003 all coupled to each other. 00:09:25.003 --> 00:09:27.290 So you get one equation for each box. 00:09:27.290 --> 00:09:30.189 And in this problem, we're going to-- 00:09:30.189 --> 00:09:31.980 when you do this method, what people assume 00:09:31.980 --> 00:09:33.646 is that you have a uniform concentration 00:09:33.646 --> 00:09:35.100 sort of across here or the number 00:09:35.100 --> 00:09:37.740 you use as the average of the concentration in the cell 00:09:37.740 --> 00:09:38.830 as [INAUDIBLE]. 00:09:38.830 --> 00:09:42.959 And so it's not really exactly realistic. 00:09:42.959 --> 00:09:44.500 So that's where the approximation is. 00:09:44.500 --> 00:09:45.720 What you're doing at the boundaries 00:09:45.720 --> 00:09:47.550 is very realistic, because you're actually computing 00:09:47.550 --> 00:09:48.930 the fluxes across control volumes 00:09:48.930 --> 00:09:50.880 is exactly the way you should do it. 00:09:50.880 --> 00:09:55.320 So there's different methods, right, finite element, 00:09:55.320 --> 00:09:57.204 finite difference, finite volume. 00:09:57.204 --> 00:09:58.620 The nice thing about finite volume 00:09:58.620 --> 00:10:00.660 is you're really treating exactly what's 00:10:00.660 --> 00:10:04.750 happening across the boundaries of the mesh area, the mesh 00:10:04.750 --> 00:10:06.079 volume. 00:10:06.079 --> 00:10:08.120 And we'll find that a lot of ChemE problems, this 00:10:08.120 --> 00:10:09.369 is how people prefer to do it. 00:10:13.380 --> 00:10:17.610 And because you're treating the fluxes exactly, 00:10:17.610 --> 00:10:26.522 if you have your mesh volume sitting on the wall 00:10:26.522 --> 00:10:28.022 or on top of the drug patch, suppose 00:10:28.022 --> 00:10:32.380 if we're sitting right on top of the drug patch here. 00:10:32.380 --> 00:10:34.600 Then if you're sitting-- 00:10:34.600 --> 00:10:35.650 first, let's do a wall. 00:10:35.650 --> 00:10:37.730 Suppose you're sitting on an impermeable wall, 00:10:37.730 --> 00:10:46.360 then we just know that the flux here will be 0 if it's a wall. 00:10:46.360 --> 00:10:47.450 So that's easy. 00:10:47.450 --> 00:10:48.247 No flux here. 00:10:48.247 --> 00:10:50.330 So we only have to worry about this one, this one, 00:10:50.330 --> 00:10:51.750 and this one. 00:10:51.750 --> 00:10:54.280 That one's not doing anything. 00:10:54.280 --> 00:10:57.170 This case, if you say in the drug patch there is a flux, 00:10:57.170 --> 00:10:59.350 stuff's coming in and you'll need 00:10:59.350 --> 00:11:02.180 to figure out how to write that boundary condition. 00:11:02.180 --> 00:11:06.100 So the boundary condition as written is that C, the drug, 00:11:06.100 --> 00:11:11.200 is equal to some number, C drug, whatever is in the patch. 00:11:11.200 --> 00:11:14.270 But that's not so easy to impose here. 00:11:14.270 --> 00:11:16.010 And so there's two ways to look at it. 00:11:16.010 --> 00:11:20.110 One way is people compute this flux by considering-- 00:11:23.225 --> 00:11:26.451 suppose I know this is C drug here. 00:11:26.451 --> 00:11:28.340 I'm trying to really figure out what's 00:11:28.340 --> 00:11:31.200 the average concentration here. 00:11:31.200 --> 00:11:33.470 And so one way to compute this is 00:11:33.470 --> 00:11:43.020 to say it's the diffusivity times C drug minus C 00:11:43.020 --> 00:11:50.620 the middle over delta y over 2. 00:11:50.620 --> 00:11:52.350 That would be the flux. 00:11:52.350 --> 00:11:54.060 So that's one way to look at it. 00:11:54.060 --> 00:11:57.830 Another way people look at it is they draw 00:11:57.830 --> 00:11:59.998 what they call a ghost volume. 00:12:03.140 --> 00:12:06.980 And so here, here's my C that I care about, C middle. 00:12:12.310 --> 00:12:13.296 Here's C ghost. 00:12:17.120 --> 00:12:18.680 And here is the line between them. 00:12:18.680 --> 00:12:20.450 Now, I can use the same equation here 00:12:20.450 --> 00:12:22.640 that they would have used before. 00:12:22.640 --> 00:12:25.730 But I don't know what C ghost is, because there's 00:12:25.730 --> 00:12:26.520 no real cell here. 00:12:26.520 --> 00:12:27.590 This is below the patch. 00:12:27.590 --> 00:12:30.350 Here's the patch. 00:12:30.350 --> 00:12:32.660 But imagine the patch is not there for a second, 00:12:32.660 --> 00:12:35.660 and I write down the same equation I would have here. 00:12:35.660 --> 00:12:41.730 And the flux diffusively would have been D C ghost minus C 00:12:41.730 --> 00:12:46.440 middle over delta y. 00:12:46.440 --> 00:12:49.220 That's what you would have written as the diffusive flux. 00:12:49.220 --> 00:12:51.285 And I don't know what C ghost is. 00:12:51.285 --> 00:12:52.660 So now, I have to think about how 00:12:52.660 --> 00:12:55.040 do I estimate what C ghost is. 00:12:55.040 --> 00:12:59.420 I can say, well, let's do a linear interpolation 00:12:59.420 --> 00:13:02.420 from this concentration to that concentration. 00:13:02.420 --> 00:13:10.520 So let's say that the C boundary is equal to C 00:13:10.520 --> 00:13:15.120 ghost, the average of these two, plus C mid over 2. 00:13:17.695 --> 00:13:19.820 That's what we got if we did a linear interpolation 00:13:19.820 --> 00:13:22.324 between these two guys to figure out what it is here. 00:13:22.324 --> 00:13:23.490 But here we know what it is. 00:13:23.490 --> 00:13:25.350 We know what a C boundary is. 00:13:25.350 --> 00:13:27.620 That's the concentration of the drug patch 00:13:27.620 --> 00:13:29.676 so that we can solve for what c ghost is. 00:13:29.676 --> 00:13:32.160 AUDIENCE: How do you know what C middle is? 00:13:32.160 --> 00:13:34.512 WILLIAM GREEN, JR: C middle is the unknown. 00:13:34.512 --> 00:13:35.970 That's what we're going to compute. 00:13:35.970 --> 00:13:38.428 We just want an equation that involves C middle, because we 00:13:38.428 --> 00:13:39.595 have c middle as an unknown. 00:13:39.595 --> 00:13:41.344 We need an equation the involves C middle, 00:13:41.344 --> 00:13:42.585 just like we would for-- 00:13:42.585 --> 00:13:43.260 if we have a-- 00:13:46.874 --> 00:13:50.050 If we have finite volume that is in the interior somewhere, 00:13:50.050 --> 00:13:53.790 we have c of this grid point, Cij. 00:13:53.790 --> 00:13:55.691 We just want equations that involve Cij. 00:13:55.691 --> 00:13:57.690 But we need an equation that somehow connects it 00:13:57.690 --> 00:13:58.620 to the boundary conditions. 00:13:58.620 --> 00:13:59.120 Yeah. 00:13:59.120 --> 00:14:01.710 AUDIENCE: When would you use that second [INAUDIBLE] 00:14:01.710 --> 00:14:04.959 if you know C boundary [INAUDIBLE]?? 00:14:04.959 --> 00:14:06.000 WILLIAM GREEN, JR: Right. 00:14:06.000 --> 00:14:08.610 And actually, I think if you do it this way, in this case, 00:14:08.610 --> 00:14:10.361 you get the same formula. 00:14:10.361 --> 00:14:11.360 You're running out of C. 00:14:11.360 --> 00:14:15.168 AUDIENCE: So when would you use the C ghost? 00:14:15.168 --> 00:14:18.230 WILLIAM GREEN, JR: The C ghost thing is 00:14:18.230 --> 00:14:22.020 handy when these conditions-- 00:14:22.020 --> 00:14:24.477 actually, in the finite volumes, this is it. 00:14:24.477 --> 00:14:25.810 I don't think you need to do it. 00:14:25.810 --> 00:14:27.260 Maybe if you had a flow, velocity flow here. 00:14:27.260 --> 00:14:27.500 I don't know. 00:14:27.500 --> 00:14:28.500 But you won't in a wall. 00:14:28.500 --> 00:14:33.780 So I don't know if, in the finite difference method, 00:14:33.780 --> 00:14:37.140 you can use those ghost points as well to do the flux gutter 00:14:37.140 --> 00:14:37.916 conditions. 00:14:37.916 --> 00:14:39.540 And that's often useful to do the ghost 00:14:39.540 --> 00:14:42.710 thing for if you have a symmetry-imposed flux boundary 00:14:42.710 --> 00:14:44.471 condition. 00:14:44.471 --> 00:14:44.970 Yeah. 00:14:44.970 --> 00:14:48.456 AUDIENCE: [INAUDIBLE] 00:14:50.362 --> 00:14:51.820 WILLIAM GREEN, JR: That's the flux. 00:14:51.820 --> 00:14:53.528 The flux is coming from across this wall. 00:14:53.528 --> 00:14:55.660 And we'll have to add that with the flux coming 00:14:55.660 --> 00:14:57.320 this way and the flux coming this way 00:14:57.320 --> 00:15:00.580 and the flux coming this way to get our total flux, which 00:15:00.580 --> 00:15:07.923 is going to net out to 0 instead of C. Is that all right? 00:15:07.923 --> 00:15:10.190 But the other ones, you all know how to write them? 00:15:10.190 --> 00:15:12.225 They're just velocity flows. 00:15:12.225 --> 00:15:14.250 The nice thing about the finite volume method 00:15:14.250 --> 00:15:22.560 is that when you're trying to consider the velocity, 00:15:22.560 --> 00:15:25.865 the velocity that matters is actually the velocity here. 00:15:25.865 --> 00:15:29.940 It's the flux that couples the two finite values together. 00:15:29.940 --> 00:15:31.860 So your velocity is being evaluated halfway 00:15:31.860 --> 00:15:36.180 between the two mesh point centers. 00:15:36.180 --> 00:15:38.760 This turns out to make the whole procedure more numerically 00:15:38.760 --> 00:15:41.250 stable. 00:15:41.250 --> 00:15:44.400 You compute the flux by, say, the difference 00:15:44.400 --> 00:15:46.130 between these two. 00:15:46.130 --> 00:15:50.660 And you're implicitly evaluating right at halfway between. 00:15:50.660 --> 00:15:53.000 Other methods, like finite difference, 00:15:53.000 --> 00:15:56.240 you're trying to get the velocity or the flux 00:15:56.240 --> 00:15:59.700 at the same point here. 00:15:59.700 --> 00:16:01.305 And it doesn't really make much sense, 00:16:01.305 --> 00:16:03.930 from the point of view of trying to compute how much material's 00:16:03.930 --> 00:16:07.210 close to here, to use the velocity right there. 00:16:07.210 --> 00:16:10.770 And so when you work out equations, and particularly 00:16:10.770 --> 00:16:13.365 the equations that involve pressure, 00:16:13.365 --> 00:16:17.340 so suppose you try to discretize the Navier-Stokes equations 00:16:17.340 --> 00:16:20.470 and you have a pressure gradient that's driving the flow, 00:16:20.470 --> 00:16:22.820 then you have to figure out where are you 00:16:22.820 --> 00:16:25.165 going to evaluate the pressures and where are you going 00:16:25.165 --> 00:16:26.120 to evaluate the velocities. 00:16:26.120 --> 00:16:28.286 And when you try to evaluate them at the same point, 00:16:28.286 --> 00:16:31.090 it turns out that you get numerically unstable problems. 00:16:31.090 --> 00:16:33.010 But if you do it by this funny volume method, 00:16:33.010 --> 00:16:34.385 then it just works out naturally. 00:16:34.385 --> 00:16:35.810 It's fine. 00:16:35.810 --> 00:16:37.490 And then you need special methods, 00:16:37.490 --> 00:16:39.554 if you're going to try to do it where you have 00:16:39.554 --> 00:16:41.970 the velocity of the mesh at the same point as the pressure 00:16:41.970 --> 00:16:42.990 mesh. 00:16:42.990 --> 00:16:44.460 You'll notice in most of the problems we give you, 00:16:44.460 --> 00:16:45.585 we just leave pressure out. 00:16:45.585 --> 00:16:48.420 We try to rewrite the equations so no pressure ever appears. 00:16:48.420 --> 00:16:50.003 And that's because pressure in general 00:16:50.003 --> 00:16:55.080 is a problem, because you can have acoustic waves physically. 00:16:55.080 --> 00:16:57.647 And if you do the equations, which allow for acoustic waves, 00:16:57.647 --> 00:16:59.730 you'll get them in the numerical solution as well. 00:16:59.730 --> 00:17:01.380 But usually, we don't care about that. 00:17:01.380 --> 00:17:04.770 So you have sound waves racing around your solution, 00:17:04.770 --> 00:17:06.849 and that causes a lot of trouble numerically. 00:17:06.849 --> 00:17:09.060 So a lot of times, people rewrite the equations 00:17:09.060 --> 00:17:12.984 to try to remove the pressure from the equation. 00:17:12.984 --> 00:17:14.900 So you write down the Navier-Stokes equations. 00:17:14.900 --> 00:17:16.807 You guys did this transport. 00:17:16.807 --> 00:17:19.140 Maybe you'll study that today in the class, in the test. 00:17:19.140 --> 00:17:19.681 I don't know. 00:17:19.681 --> 00:17:22.790 Anyway, there's a pressure term if you just write naturally. 00:17:22.790 --> 00:17:25.910 But oftentimes, you can remove that by an equation of state, 00:17:25.910 --> 00:17:26.780 for example. 00:17:26.780 --> 00:17:28.470 And then you can get rid of it. 00:17:28.470 --> 00:17:31.932 And that turns out to be better from the numerical solution. 00:17:31.932 --> 00:17:33.380 All right. 00:17:33.380 --> 00:17:33.880 Yes. 00:17:33.880 --> 00:17:35.060 AUDIENCE: I have a question. 00:17:35.060 --> 00:17:37.410 So you said that for this problem, 00:17:37.410 --> 00:17:39.165 [INAUDIBLE] was very small. 00:17:39.165 --> 00:17:44.667 So that means that these finite volume cells [INAUDIBLE] you 00:17:44.667 --> 00:17:45.950 have to do a lot of them. 00:17:45.950 --> 00:17:49.648 [INAUDIBLE] gets around this by having a variable volume. 00:17:49.648 --> 00:17:52.681 Do we have to address this on that lab that 00:17:52.681 --> 00:17:55.396 have some variable volume? 00:17:55.396 --> 00:17:57.190 WILLIAM GREEN, JR: OK, so from the point 00:17:57.190 --> 00:18:00.570 of view of writing a code, it's a lot easier 00:18:00.570 --> 00:18:03.720 to write it with fixed mesh because all equations 00:18:03.720 --> 00:18:05.200 look exactly the same. 00:18:05.200 --> 00:18:08.320 So I suggest you start that way. 00:18:08.320 --> 00:18:13.160 And what you do, if you use a very small value of each, 00:18:13.160 --> 00:18:16.090 then I think you'll be able solve it, no problem. 00:18:16.090 --> 00:18:18.260 You'll have enough mesh. 00:18:18.260 --> 00:18:21.390 Then, once you figure out how to do 00:18:21.390 --> 00:18:23.340 that, now you might be able to see, 00:18:23.340 --> 00:18:25.116 OK, can my solver solve this? 00:18:25.116 --> 00:18:27.240 And then that gets into another question, actually. 00:18:27.240 --> 00:18:29.910 What solver are you going to use? 00:18:29.910 --> 00:18:34.760 So any ideas about this problem? 00:18:34.760 --> 00:18:37.170 AUDIENCE: [INAUDIBLE] 00:18:37.170 --> 00:18:40.380 WILLIAM GREEN, JR: So you could use fsolve But I'm telling you, 00:18:40.380 --> 00:18:41.480 you're going to have to use a lot of mesh points. 00:18:41.480 --> 00:18:42.740 That means that fsolve's going to have to solve 00:18:42.740 --> 00:18:43.698 for a lot of variables. 00:18:43.698 --> 00:18:45.770 AUDIENCE: 00:18:45.770 --> 00:18:47.800 WILLIAM GREEN, JR: Backslash it might be. 00:18:47.800 --> 00:18:52.920 So the key thing, the nice thing about this problem 00:18:52.920 --> 00:18:56.030 is it's a linear differential equation. 00:18:56.030 --> 00:18:59.040 There's no nonlinear term today. 00:18:59.040 --> 00:19:01.670 So when you rewrite the equations 00:19:01.670 --> 00:19:04.290 of the finite volumes, it's all going 00:19:04.290 --> 00:19:07.890 to be linear in the unknowns. 00:19:07.890 --> 00:19:10.260 And what else is nice about this problem, when you use 00:19:10.260 --> 00:19:12.720 local finite volumes as your-- 00:19:12.720 --> 00:19:14.340 It's going to be super-sparse. 00:19:14.340 --> 00:19:17.922 So the matrix that comes in is going to be really sparse. 00:19:17.922 --> 00:19:20.130 And so you'll want to use some method that can handle 00:19:20.130 --> 00:19:23.610 gigantic sparse matrices. 00:19:23.610 --> 00:19:25.226 So you wrote a code like that earlier, 00:19:25.226 --> 00:19:26.600 so that would be one possibility. 00:19:26.600 --> 00:19:29.550 If you know how to use the right flags in MATLAB 00:19:29.550 --> 00:19:32.559 to help their built-in solvers handle sparsity, 00:19:32.559 --> 00:19:33.600 then that should be good. 00:19:33.600 --> 00:19:35.558 If you just ask it to solve it by dense method, 00:19:35.558 --> 00:19:37.340 by, like, LEU or something, you're 00:19:37.340 --> 00:19:38.923 going to have lot trouble, once you've 00:19:38.923 --> 00:19:41.040 put your mesh points in there. 00:19:41.040 --> 00:19:42.240 But you can just experiment. 00:19:42.240 --> 00:19:43.260 Try bigger and bigger matrices. 00:19:43.260 --> 00:19:45.265 And then at some point, if you have backslash, 00:19:45.265 --> 00:19:48.530 it will give you a warning or something, unless you tell it 00:19:48.530 --> 00:19:51.842 that you're sparse. 00:19:51.842 --> 00:19:53.674 All right? 00:19:53.674 --> 00:19:56.100 Any more questions that about this? 00:19:56.100 --> 00:19:58.350 How would people solve it do you think professionally? 00:19:58.350 --> 00:20:01.747 Suppose I was doing a problem like this in 3D instead of 2D. 00:20:01.747 --> 00:20:02.830 How would people solve it? 00:20:02.830 --> 00:20:03.955 What solver would they use? 00:20:03.955 --> 00:20:05.881 AUDIENCE: [INAUDIBLE] 00:20:05.881 --> 00:20:07.130 WILLIAM GREEN, JR: Not fsolve. 00:20:07.130 --> 00:20:07.745 No. 00:20:07.745 --> 00:20:09.417 AUDIENCE: [INAUDIBLE] gradient. 00:20:09.417 --> 00:20:12.000 WILLIAM GREEN, JR: Yeah, so they would use conjugate gradient. 00:20:12.000 --> 00:20:18.410 So probably-- I think it's called this, BiCGSTAB. 00:20:18.410 --> 00:20:19.940 That's the program that we would use 00:20:19.940 --> 00:20:23.570 if you have a really, really gigantic sparse matrix. 00:20:23.570 --> 00:20:25.370 So that's the conjugate gradient. 00:20:25.370 --> 00:20:29.459 And Professor Swan talked about, the advantage 00:20:29.459 --> 00:20:31.250 of that is you never have to actually store 00:20:31.250 --> 00:20:33.050 the whole matrix. 00:20:33.050 --> 00:20:35.390 You only need to evaluate the matrix elements, 00:20:35.390 --> 00:20:36.914 and then you can throw them away. 00:20:36.914 --> 00:20:38.580 And so if you have a very sparse matrix, 00:20:38.580 --> 00:20:40.360 that's pretty cheap to do. 00:20:40.360 --> 00:20:42.507 So it's a really good code. 00:20:42.507 --> 00:20:44.340 I think in this 2D problem, you can probably 00:20:44.340 --> 00:20:45.270 get away with other solvers. 00:20:45.270 --> 00:20:45.960 You don't have to use this. 00:20:45.960 --> 00:20:47.418 But this is a definite possibility. 00:20:47.418 --> 00:20:49.560 This is a built-in MATLAB program as well. 00:20:49.560 --> 00:20:53.310 Be warned, though, this is an iterative solver. 00:20:53.310 --> 00:20:56.005 It's not just going to be one solve, boom. 00:20:56.005 --> 00:20:57.130 And It might have troubles. 00:20:59.660 --> 00:21:04.330 So you might want to go with the other ones, but anyway. 00:21:04.330 --> 00:21:05.540 May I ask you a question? 00:21:05.540 --> 00:21:07.081 How about, do you need initial guess? 00:21:12.133 --> 00:21:12.930 What do you think? 00:21:12.930 --> 00:21:13.702 AUDIENCE: Depends. 00:21:13.702 --> 00:21:15.410 WILLIAM GREEN, JR: Depends on the solver. 00:21:15.410 --> 00:21:16.960 So if you solve it with backslash, 00:21:16.960 --> 00:21:19.070 do you need an initial guess? 00:21:19.070 --> 00:21:22.610 If you can solve it with fsolve, do you have to initial guess? 00:21:22.610 --> 00:21:24.230 If you can solve it via BiCGSTAB, 00:21:24.230 --> 00:21:26.532 do you have to use initial guess? 00:21:26.532 --> 00:21:29.328 AUDIENCE: Yes. 00:21:29.328 --> 00:21:30.871 WILLIAM GREEN, JR: Yes. 00:21:30.871 --> 00:21:31.370 OK. 00:21:31.370 --> 00:21:33.510 So then you have to think about how you really 00:21:33.510 --> 00:21:34.676 get your initial guess, too. 00:21:34.676 --> 00:21:39.010 So this is things to think about. 00:21:39.010 --> 00:21:39.510 All right. 00:21:43.680 --> 00:21:45.730 What else to tell you about? 00:21:45.730 --> 00:21:50.120 One last thing about PDEs, and we'll 00:21:50.120 --> 00:21:54.020 come back to this later, so far we haven't done really 00:21:54.020 --> 00:21:56.330 anything that's very time-dependent. 00:21:56.330 --> 00:22:00.290 But a lot of real world PDEs have a time dependent in them. 00:22:00.290 --> 00:22:02.160 And there's is a very important concept, 00:22:02.160 --> 00:22:03.470 a thing called the CFL number. 00:22:06.230 --> 00:22:09.230 And this is named after a 1928 paper, 00:22:09.230 --> 00:22:10.730 and I'll write the guys' names down. 00:22:21.090 --> 00:22:23.700 And what they showed was that, if you're 00:22:23.700 --> 00:22:27.300 trying to solve the PDE system, where 00:22:27.300 --> 00:22:32.470 you're discretizing in both x and time, 00:22:32.470 --> 00:22:35.820 that you have a number that they defined 00:22:35.820 --> 00:22:39.630 as delta t times the velocity and the extraction 00:22:39.630 --> 00:22:41.920 divided by delta x. 00:22:41.920 --> 00:22:43.810 So that's a dimensionless number. 00:22:43.810 --> 00:22:46.560 So that's a CFL number. 00:22:46.560 --> 00:22:47.940 You see a lot of papers that will 00:22:47.940 --> 00:22:49.800 say what CFL number they used. 00:22:49.800 --> 00:22:52.620 What that means is the ratio of their time mesh compared 00:22:52.620 --> 00:22:54.850 to their space mesh. 00:22:54.850 --> 00:23:00.020 And conceptually, let's think about what's happening here. 00:23:00.020 --> 00:23:02.020 So suppose we have a flow flowing 00:23:02.020 --> 00:23:06.610 in an upwards direction, and we have a bunch 00:23:06.610 --> 00:23:08.794 of little finite volumes. 00:23:11.960 --> 00:23:14.780 So we've discretized the delta x already. 00:23:14.780 --> 00:23:18.250 And this is x. 00:23:18.250 --> 00:23:20.770 And there's a flow here. 00:23:20.770 --> 00:23:23.207 And I've already decided, somehow, 00:23:23.207 --> 00:23:24.540 what length scale I want to use. 00:23:24.540 --> 00:23:26.150 So I've decided my delta x. 00:23:26.150 --> 00:23:29.430 And now, I'm trying to figure out what delta t I should use. 00:23:29.430 --> 00:23:32.180 Now, from the point of view of saving CPU time 00:23:32.180 --> 00:23:34.670 I want the delta to be as giant as possible, 00:23:34.670 --> 00:23:37.400 because I want to be able to zoom along or predict 00:23:37.400 --> 00:23:40.350 for long periods of time what's going to happen in my system. 00:23:40.350 --> 00:23:42.009 But if I make delta t really large, 00:23:42.009 --> 00:23:43.300 let's think about what happens. 00:23:43.300 --> 00:23:51.980 Suppose I choose delta t to be 10 times delta x divided by u. 00:23:51.980 --> 00:23:53.940 So it means that in my one time step, 00:23:53.940 --> 00:23:57.289 I have some guess or some current value 00:23:57.289 --> 00:23:59.330 of the concentrations in all these finite values. 00:24:01.840 --> 00:24:08.960 And then I wait through a time step that's 00:24:08.960 --> 00:24:12.320 10 times delta x over ux. 00:24:12.320 --> 00:24:14.830 So I had some stuff that was here. 00:24:14.830 --> 00:24:18.320 Where is that going to be 10 times later? 00:24:18.320 --> 00:24:20.877 10 time steps later. 00:24:20.877 --> 00:24:21.710 10 blocks up, right? 00:24:21.710 --> 00:24:26.140 So it's going to be, like, way up here somewhere. 00:24:26.140 --> 00:24:29.680 And so what's going to happen there 00:24:29.680 --> 00:24:34.980 is that my numerical methods are all computing stuff locally 00:24:34.980 --> 00:24:36.860 from the spatial derivatives. 00:24:36.860 --> 00:24:39.860 But it's crazy if, between my time steps, 00:24:39.860 --> 00:24:42.140 this stuff completely left the picture. 00:24:42.140 --> 00:24:44.726 It's already convected all the way off the screen. 00:24:44.726 --> 00:24:46.850 And some new stuff, which was way down here before, 00:24:46.850 --> 00:24:48.624 is now the stuff that's here. 00:24:48.624 --> 00:24:50.880 Should be there if I was physical. 00:24:50.880 --> 00:24:54.120 Numerically, who knows what will happen if you try to do this. 00:24:54.120 --> 00:24:56.130 But it won't be good. 00:24:56.130 --> 00:24:59.040 So the condition is that you need 00:24:59.040 --> 00:25:03.060 this number to be less than 1, or same order of magnitude 00:25:03.060 --> 00:25:06.081 as 1, and you try to make this much bigger than 1, 00:25:06.081 --> 00:25:08.080 then you're doing something crazy because you're 00:25:08.080 --> 00:25:13.210 convecting stuff over multiple mesh points. 00:25:13.210 --> 00:25:16.649 And so that turns out to be a very serious limitation if you 00:25:16.649 --> 00:25:18.190 try to do simulations for, let's say, 00:25:18.190 --> 00:25:21.070 a reacting flow for a long period of time, 00:25:21.070 --> 00:25:23.480 because you might have to use a really tiny delta t. 00:25:23.480 --> 00:25:27.010 And then people have developed all different fancy methods 00:25:27.010 --> 00:25:28.720 to try to get around that. 00:25:28.720 --> 00:25:32.170 But if you just do the obvious things to do, 00:25:32.170 --> 00:25:33.805 you'll always run into this limitation. 00:25:33.805 --> 00:25:37.970 Then you need to choose the time steps small. 00:25:37.970 --> 00:25:43.190 And also, it's bad, because as you make delta x smaller, which 00:25:43.190 --> 00:25:45.780 improves your accuracy, You'll have to make 00:25:45.780 --> 00:25:47.720 your delta t's smaller, too. 00:25:47.720 --> 00:25:50.690 But of course, making delta x smaller increases your CPU time 00:25:50.690 --> 00:25:53.360 because you have more finite volumes to compute. 00:25:53.360 --> 00:25:55.040 And then you'll also have to make double t smaller, which 00:25:55.040 --> 00:25:55.910 means you'll have to do more time steps, too. 00:25:55.910 --> 00:25:58.040 So it's even like a double whammy. 00:25:58.040 --> 00:26:01.400 So getting more accuracy is going to really cost you badly. 00:26:01.400 --> 00:26:05.520 And so this another reason why people used to always 00:26:05.520 --> 00:26:06.862 refer to color fluid dynamics. 00:26:06.862 --> 00:26:08.820 You can make a pretty picture, but it might not 00:26:08.820 --> 00:26:09.935 be physical at all. 00:26:09.935 --> 00:26:12.060 Because you can make it solve equations that maybe, 00:26:12.060 --> 00:26:13.470 for example, didn't impose this, then 00:26:13.470 --> 00:26:14.970 who knows what kind of crazy stuff you'll get. 00:26:14.970 --> 00:26:16.050 You'll got something. 00:26:16.050 --> 00:26:18.000 It'll compute something, but it may have no relation 00:26:18.000 --> 00:26:18.833 to the real problem. 00:26:23.104 --> 00:26:24.770 I think that's all I was say about PDEs. 00:26:24.770 --> 00:26:26.811 Are there any questions about PDEs before I start 00:26:26.811 --> 00:26:28.128 talking about probability? 00:26:30.811 --> 00:26:31.810 You got it totally down. 00:26:31.810 --> 00:26:34.018 I'm looking forward to some really awesome solutions. 00:26:34.018 --> 00:26:34.887 How about that? 00:26:44.940 --> 00:26:46.090 Just one last comment. 00:26:46.090 --> 00:26:48.550 If you decide you wanted to do adaptive meshing 00:26:48.550 --> 00:26:53.380 and you want to change your mesh size, 00:26:53.380 --> 00:26:56.369 you can choose measures like this if you want. 00:26:56.369 --> 00:26:57.910 And you can even do things like this, 00:26:57.910 --> 00:26:58.940 where you have a bigger mesh, and then 00:26:58.940 --> 00:27:00.731 maybe have two smaller meshes underneath it 00:27:00.731 --> 00:27:03.040 in the next trial. 00:27:03.040 --> 00:27:05.120 So you can just have stuff flowing here, 00:27:05.120 --> 00:27:08.557 stuff flowing here stuff flowing there. 00:27:08.557 --> 00:27:10.390 So you can do all kinds of crazy stuff like. 00:27:10.390 --> 00:27:13.870 This can really help improve the accuracy of the solution a lot. 00:27:13.870 --> 00:27:16.300 But it's, I would say, very prone to bugs. 00:27:16.300 --> 00:27:18.220 So if you do this, be really careful 00:27:18.220 --> 00:27:19.240 and don't do it too often, I would say. 00:27:19.240 --> 00:27:20.260 You might have a few boundaries where 00:27:20.260 --> 00:27:22.870 you do something funky like that to change that mesh size. 00:27:22.870 --> 00:27:24.480 But don't go crazy with it. 00:27:24.480 --> 00:27:25.650 [INAUDIBLE] is smart. 00:27:25.650 --> 00:27:27.900 It has a really nice way of doing the meshing for you. 00:27:27.900 --> 00:27:32.170 So that's its advantage, that somebody very carefully coded 00:27:32.170 --> 00:27:34.660 how to handle this kind of stuff in a general case. 00:27:34.660 --> 00:27:36.160 But if you guys are doing it for the first time, 00:27:36.160 --> 00:27:37.260 it might not be so good. 00:27:41.330 --> 00:27:42.590 So that's enough of PDEs. 00:27:42.590 --> 00:27:45.040 Let's talk about probability. 00:28:09.040 --> 00:28:13.750 So probability is everywhere, except in undergraduate ChemE 00:28:13.750 --> 00:28:14.500 homework problems. 00:28:17.200 --> 00:28:20.142 So when you do problems as an undergraduate, 00:28:20.142 --> 00:28:21.600 they always had some nice solution. 00:28:21.600 --> 00:28:25.050 It was 2 pi, it was 3.0. 00:28:25.050 --> 00:28:27.580 Everything was, like, deterministic, is definite. 00:28:27.580 --> 00:28:29.182 The grader could go through. 00:28:29.182 --> 00:28:30.265 Oh, no, you're off by 0.1. 00:28:30.265 --> 00:28:33.820 It couldn't possibly be right. 00:28:33.820 --> 00:28:35.050 That's, like, not reality. 00:28:35.050 --> 00:28:37.300 So any time you actually make a measurement, 00:28:37.300 --> 00:28:38.855 you always had measurement noise. 00:28:38.855 --> 00:28:40.480 And if you try to repeat a measurement, 00:28:40.480 --> 00:28:42.340 you don't get the same result. So that's, 00:28:42.340 --> 00:28:45.630 like, completely different than an undergraduate problem. 00:28:45.630 --> 00:28:47.120 But this is the reality. 00:28:47.120 --> 00:28:51.480 So the reality is the world is, like, not so nice as you think. 00:28:51.480 --> 00:28:53.191 But actually, it's even more fundamental. 00:28:53.191 --> 00:28:54.690 It's nothing about-- I mean, there's 00:28:54.690 --> 00:28:56.440 one problem about how good an experimental 00:28:56.440 --> 00:28:59.550 those people are, and how fancy an apparatus 00:28:59.550 --> 00:29:03.240 you bought that can make things exactly reproducible. 00:29:03.240 --> 00:29:06.720 But even if you do that perfectly, the equations we 00:29:06.720 --> 00:29:10.590 use really don't correspond to the real physical reality. 00:29:10.590 --> 00:29:12.765 So we always use the continuum equations. 00:29:12.765 --> 00:29:16.680 You guys probably are studying them a lot in 1040 and 1050, 00:29:16.680 --> 00:29:19.314 especially 1050, I guess. 00:29:19.314 --> 00:29:20.730 But those equations are really all 00:29:20.730 --> 00:29:24.330 derived from averages over ensembles 00:29:24.330 --> 00:29:26.190 or little finite volumes or something 00:29:26.190 --> 00:29:29.760 if you look at the derivation of the equations. 00:29:29.760 --> 00:29:33.830 And reality is that the world is full of molecules. 00:29:33.830 --> 00:29:38.160 And so they're all wiggling around. 00:29:38.160 --> 00:29:39.930 And if you look in a little finite volume, 00:29:39.930 --> 00:29:41.280 you look right now, you'll see that there's 00:29:41.280 --> 00:29:42.520 27 molecules in there. 00:29:42.520 --> 00:29:44.690 If you look a second later, there might be 28. 00:29:44.690 --> 00:29:46.770 Then a little later, maybe 26. 00:29:46.770 --> 00:29:49.770 It's always fluctuating around. 00:29:49.770 --> 00:29:52.860 But according to our average equations that we use, 00:29:52.860 --> 00:29:56.805 like in the Navier-Stokes equations, it always 27.3. 00:29:56.805 --> 00:29:58.680 But of course, there's not 0.3 of a molecule. 00:29:58.680 --> 00:30:00.980 So, I mean, explicitly, it's the average 00:30:00.980 --> 00:30:03.750 is what we're computing, and the reality 00:30:03.750 --> 00:30:06.520 is fluctuating around the edges. 00:30:06.520 --> 00:30:08.250 Same thing in thermo. 00:30:08.250 --> 00:30:11.490 We say that such-and-such has a certain amount of energy. 00:30:11.490 --> 00:30:14.970 But you guys have some [INAUDIBLE] already, yes? 00:30:14.970 --> 00:30:17.250 So you saw that's not true? 00:30:17.250 --> 00:30:19.180 So really, all that's saying is that's, like, 00:30:19.180 --> 00:30:20.660 the probability, the average. 00:30:20.660 --> 00:30:24.112 If you've had many, many ensembles that 00:30:24.112 --> 00:30:26.070 were exactly the same and you average them all, 00:30:26.070 --> 00:30:28.194 you get some number and that's your average energy. 00:30:28.194 --> 00:30:30.105 But for any actual realization, it 00:30:30.105 --> 00:30:31.730 has some different value of the energy. 00:30:35.380 --> 00:30:37.680 And it's even worse than that. 00:30:37.680 --> 00:30:40.362 That's because we have a lot of particles. 00:30:40.362 --> 00:30:42.570 You can even go down to, like, the microscopic level, 00:30:42.570 --> 00:30:44.629 where you have one molecule. 00:30:44.629 --> 00:30:46.670 And you calculate things about that, it turns out 00:30:46.670 --> 00:30:48.340 you have to use the Schrodinger equation for that. 00:30:48.340 --> 00:30:49.480 And the Schrodinger equation explicitly 00:30:49.480 --> 00:30:50.725 only gives you probability densities. 00:30:50.725 --> 00:30:52.270 So it just tells you the probability 00:30:52.270 --> 00:30:53.800 the molecule might be somewhere, the electron 00:30:53.800 --> 00:30:56.110 might be somewhere, the energy might be something. 00:30:56.110 --> 00:30:58.240 But it's not actually whether it really is. 00:30:58.240 --> 00:31:00.430 It's just saying a probability distribution. 00:31:00.430 --> 00:31:02.638 And every time you do the experiment on the molecule, 00:31:02.638 --> 00:31:04.090 you get a different result. 00:31:04.090 --> 00:31:07.190 Now, this is super-annoying, but it's the way life is. 00:31:07.190 --> 00:31:09.190 Einstein got so annoyed, he has a famous saying, 00:31:09.190 --> 00:31:10.606 and it's "God does not play dice." 00:31:10.606 --> 00:31:13.250 He was, like, just completely annoyed at these equations. 00:31:13.250 --> 00:31:16.300 But it's the way it is. 00:31:16.300 --> 00:31:20.980 So the reality is that things, all we know about, really, 00:31:20.980 --> 00:31:22.500 are probability distributions. 00:31:22.500 --> 00:31:24.730 In most of our work, in our lives, 00:31:24.730 --> 00:31:27.799 we always talk about, like, the mean or the median. 00:31:27.799 --> 00:31:29.590 And we're talking as if it's a real number. 00:31:29.590 --> 00:31:32.470 But really, it's always some distribution. 00:31:32.470 --> 00:31:34.840 So it's time, I guess, you're in graduate school, 00:31:34.840 --> 00:31:36.680 it's time to, like, face up to this. 00:31:36.680 --> 00:31:38.138 And that's what we're going to talk 00:31:38.138 --> 00:31:39.826 about for the next week or two. 00:31:43.695 --> 00:31:46.774 AUDIENCE: [INAUDIBLE] 00:31:46.774 --> 00:31:48.190 WILLIAM GREEN, JR: This also makes 00:31:48.190 --> 00:31:49.270 it makes you wonder what you're doing 00:31:49.270 --> 00:31:50.600 when you make a measurement. 00:31:50.600 --> 00:31:53.769 So if you make a measurement, first of all, the fact 00:31:53.769 --> 00:31:55.560 that when you measure something repeatedly, 00:31:55.560 --> 00:31:58.270 you're not going to get the same number, that's alarming. 00:31:58.270 --> 00:32:00.640 Because I want to say I'm 5 foot 9", 00:32:00.640 --> 00:32:03.305 this should be pretty common, I'm always 5 foot 9". 00:32:03.305 --> 00:32:05.680 But actually, if you measure me multiple times, sometimes 00:32:05.680 --> 00:32:07.870 you'll get 5' 9 and 1/4" Some people will get 5' 00:32:07.870 --> 00:32:13.450 8 and 3/4" So it might make you worry, did I change? 00:32:13.450 --> 00:32:17.620 Did I grow between the measurements? 00:32:17.620 --> 00:32:19.300 So that's one issue. 00:32:23.680 --> 00:32:31.035 Because our experiments are not repeatable 00:32:31.035 --> 00:32:32.660 and we get different numbers every time 00:32:32.660 --> 00:32:35.414 we make a measurement, then we have a big problem. 00:32:35.414 --> 00:32:36.830 Somebody says, well, I really want 00:32:36.830 --> 00:32:38.720 to know how tall Professor Green is. 00:32:38.720 --> 00:32:41.299 I've always wondered, how tall is he? 00:32:41.299 --> 00:32:43.090 And everybody's told me a different number. 00:32:43.090 --> 00:32:44.465 And when you go measure it again, 00:32:44.465 --> 00:32:45.860 you get a different number again. 00:32:45.860 --> 00:32:48.380 What's going on? 00:32:48.380 --> 00:32:49.970 And so you'll have to then-- then we 00:32:49.970 --> 00:32:51.740 have, like, a concept that there is 00:32:51.740 --> 00:32:54.170 a true height of Professor Green and we just 00:32:54.170 --> 00:32:56.160 don't know what it is. 00:32:56.160 --> 00:32:57.950 And then we'll try maybe to make repeated 00:32:57.950 --> 00:33:01.637 measurements of my height, and then maybe take the average. 00:33:01.637 --> 00:33:03.220 That would be the obvious thing to do. 00:33:03.220 --> 00:33:04.952 And we take the average and report that. 00:33:04.952 --> 00:33:06.910 We'll tell the boss, we'll lie to him, say, oh, 00:33:06.910 --> 00:33:08.395 Professor Green's 5 foot 9". 00:33:08.395 --> 00:33:10.520 When really, we never actually measured 5' foot 9". 00:33:10.520 --> 00:33:14.355 Every time we measured, it was 5' 9 and 1/8", 5' 8 and 3/4". 00:33:14.355 --> 00:33:15.980 Every time, it was something different. 00:33:15.980 --> 00:33:17.470 But we just say, OK, it's 5' 9". 00:33:17.470 --> 00:33:19.400 And the boss, he doesn't want to know about all this complexity, 00:33:19.400 --> 00:33:20.733 anyway, so he just believes you. 00:33:20.733 --> 00:33:22.460 So it takes your average number. 00:33:22.460 --> 00:33:27.290 But you know that you're not really sure I'm exactly 5' 9" 00:33:27.290 --> 00:33:30.560 And so you have a probability distribution, 00:33:30.560 --> 00:33:32.280 and you're doing your best guess. 00:33:32.280 --> 00:33:34.310 And in fact, if you're honest to your boss, 00:33:34.310 --> 00:33:35.170 you'll give him an error bar. 00:33:35.170 --> 00:33:37.211 You'll say he's 5' 9", plus or minus 1/2 an inch. 00:33:39.980 --> 00:33:41.657 And that way, what you're saying is 00:33:41.657 --> 00:33:43.865 you're pretty sure the true height of Professor Green 00:33:43.865 --> 00:33:44.990 is somewhere in that range. 00:33:47.340 --> 00:33:48.630 Is this OK? 00:33:48.630 --> 00:33:49.260 Yeah. 00:33:49.260 --> 00:33:52.760 Now it might be that I'm actually changing height. 00:33:52.760 --> 00:33:55.010 I get a good night's rest, I lie down for a long time. 00:33:55.010 --> 00:33:55.820 Maybe when I stretch out a little bit. 00:33:55.820 --> 00:33:58.195 When I stand up in front of lecture here for a long time, 00:33:58.195 --> 00:33:59.005 I'm shrinking. 00:33:59.005 --> 00:34:00.380 My vertebrae are being compressed 00:34:00.380 --> 00:34:01.421 by standing here so long. 00:34:01.421 --> 00:34:03.680 So it's, like, a combination of things. 00:34:03.680 --> 00:34:05.870 One is your measurement system is not perfect, 00:34:05.870 --> 00:34:09.040 and one is that I actually might be fluctuating. 00:34:09.040 --> 00:34:10.800 I had a big breakfast, I'm growing. 00:34:13.310 --> 00:34:14.859 So is it true with everything? 00:34:14.859 --> 00:34:16.400 Every experiment you do is like this, 00:34:16.400 --> 00:34:19.639 that there is a real fluctuation of the real, physical apparatus 00:34:19.639 --> 00:34:22.340 of the thing that you're trying to measure because mostly, 00:34:22.340 --> 00:34:24.340 things we're trying to measure have fluctuations 00:34:24.340 --> 00:34:25.159 intrinsically. 00:34:25.159 --> 00:34:28.570 And then on top of that, your measurement device 00:34:28.570 --> 00:34:30.590 is fluctuating, which means-- and then 00:34:30.590 --> 00:34:32.298 the combination is what you're measuring. 00:34:32.298 --> 00:34:35.270 It gives you fluctuation. 00:34:35.270 --> 00:34:36.949 If you have a very nice instrument, 00:34:36.949 --> 00:34:38.719 the fluctuation of your instrument 00:34:38.719 --> 00:34:41.239 is smaller than the fluctuation in the real system 00:34:41.239 --> 00:34:42.830 and you'll go down to the limit. 00:34:42.830 --> 00:34:44.000 Anybody with a really good instrument 00:34:44.000 --> 00:34:45.920 should measure approximately the same probability distribution, 00:34:45.920 --> 00:34:47.544 which is actually the real fluctuation. 00:34:47.544 --> 00:34:48.760 My height, for example. 00:34:48.760 --> 00:34:50.385 So if you bought a laser interferometer 00:34:50.385 --> 00:34:53.389 and mounted a mirror on my head, and measured my height 00:34:53.389 --> 00:34:55.130 to the wavelength of light, you're 00:34:55.130 --> 00:34:57.080 pretty sure that it's pretty good. 00:34:57.080 --> 00:34:58.880 It's within a wavelength of light. 00:34:58.880 --> 00:35:00.796 So the error bar there is just due to the fact 00:35:00.796 --> 00:35:02.620 that I slouch sometimes. 00:35:02.620 --> 00:35:03.450 Is that OK? 00:35:07.300 --> 00:35:09.650 So let's talk about some basic things about probability. 00:35:09.650 --> 00:35:13.850 So we're always saying there's a probability of an event. 00:35:19.710 --> 00:35:21.500 And so we want to give a number. 00:35:21.500 --> 00:35:23.570 So for example, I'm flipping coins. 00:35:23.570 --> 00:35:26.360 I flip a penny, it could be heads or tails. 00:35:26.360 --> 00:35:31.902 I'll say the probability of heads 00:35:31.902 --> 00:35:35.950 is approximately equal to 1/2. 00:35:35.950 --> 00:35:42.230 So I flipped the coin, and if you flip the coin 100 times, 00:35:42.230 --> 00:35:43.880 you might expect to see 50 heads. 00:35:43.880 --> 00:35:47.600 Now, it could be 49 heads, it could be 51 heads. 00:35:47.600 --> 00:35:50.060 So you have to worry about, like, exactly how precisely you 00:35:50.060 --> 00:35:50.560 know it. 00:35:50.560 --> 00:35:52.750 But you think it's something like a half. 00:35:52.750 --> 00:35:55.700 Now, just to warn you, I actually 00:35:55.700 --> 00:35:58.830 didn't specify any more significant figures here, 00:35:58.830 --> 00:36:00.750 and it might be really hard to figure out what 00:36:00.750 --> 00:36:02.312 those significant figures are. 00:36:02.312 --> 00:36:04.270 And this is related to the measurement problem. 00:36:04.270 --> 00:36:07.620 So we think that a coin has about a 50/50 chance 00:36:07.620 --> 00:36:08.880 of being heads or tails. 00:36:08.880 --> 00:36:10.700 But if you really wanted to prove it, 00:36:10.700 --> 00:36:13.090 that might be really hard to do. 00:36:13.090 --> 00:36:15.268 You can have joint probabilities. 00:36:20.890 --> 00:36:27.640 So suppose I have a penny and I have a dime. 00:36:27.640 --> 00:36:30.430 And I try to think, if I flipped them both, what could happen? 00:36:30.430 --> 00:36:33.194 I could have that they both come up heads. 00:36:33.194 --> 00:36:35.360 I could have this guy come up heads, this guy tails. 00:36:35.360 --> 00:36:38.470 This guy tails, this heads, tails. 00:36:38.470 --> 00:36:40.504 So there's, like, four possible outcomes, 00:36:40.504 --> 00:36:42.420 and we think they all have about approximately 00:36:42.420 --> 00:36:43.815 equal probabilities. 00:36:43.815 --> 00:36:45.940 So the probability of any of these things happening 00:36:45.940 --> 00:36:46.773 should be about 1/4. 00:36:49.900 --> 00:36:59.560 So you can write the probability of event 1 and event 2. 00:37:02.320 --> 00:37:04.330 So this would be, for example, the probability 00:37:04.330 --> 00:37:08.180 that I got heads for penny and heads for dime, 00:37:08.180 --> 00:37:09.850 and I think that this is about 1/4. 00:37:19.276 --> 00:37:32.802 Now, I could also say the probability of heads, 00:37:32.802 --> 00:37:37.620 and heads is equal to the probability of heads 00:37:37.620 --> 00:37:45.090 for the penny times the probability of heads 00:37:45.090 --> 00:37:53.090 for the dime if I got heads for the penny. 00:37:53.090 --> 00:37:58.860 Now, if these two coin flips are completely uncorrelated, 00:37:58.860 --> 00:38:02.310 then the probability of heads on the dime and heads on the penny 00:38:02.310 --> 00:38:02.830 is the same. 00:38:02.830 --> 00:38:04.955 It's just the probability of the heads on the dime. 00:38:04.955 --> 00:38:06.390 So they don't matter. 00:38:06.390 --> 00:38:09.992 But many things that we'll study are correlated. 00:38:09.992 --> 00:38:11.700 That the probability of something happens 00:38:11.700 --> 00:38:13.650 depends on whether something else happened. 00:38:17.320 --> 00:38:19.780 So this kind of expression is very important. 00:38:19.780 --> 00:38:21.080 Now, this is just an equality. 00:38:21.080 --> 00:38:24.070 It's like a definition of what these are, right? 00:38:24.070 --> 00:38:26.320 And I'll just notice you can write that the other way 00:38:26.320 --> 00:38:26.830 around. 00:38:26.830 --> 00:38:31.550 So it's the probability of heads on the dime 00:38:31.550 --> 00:38:34.780 times the probability of heads on the penny. 00:38:41.486 --> 00:38:43.950 This is OK? 00:38:43.950 --> 00:38:46.250 So these two guys are equal to each other, 00:38:46.250 --> 00:38:48.860 and you can rearrange that equation any way you want. 00:38:48.860 --> 00:38:50.570 And we'll come back to a very famous way 00:38:50.570 --> 00:38:51.611 to rewrite that equation. 00:38:51.611 --> 00:38:54.320 It's called Bayes' theorem, and that turns out 00:38:54.320 --> 00:38:57.486 to be really important in model versus data comparisons. 00:39:02.450 --> 00:39:07.100 Instead of doing AND, do OR. 00:39:07.100 --> 00:39:10.420 So maybe you can say, then, what's the probability 00:39:10.420 --> 00:39:13.350 that I see at least one head? 00:39:19.240 --> 00:39:22.910 So I flip my two coins, and I have the probability of, 00:39:22.910 --> 00:39:24.382 I see at least one head. 00:39:28.520 --> 00:39:30.935 So we know we intuitively the answer is 3/4. 00:39:33.500 --> 00:39:36.260 But let's try to think of where does that really come from. 00:39:41.340 --> 00:39:42.684 So-- 00:39:42.684 --> 00:39:44.492 [HIGH-PITCHED SOUND] 00:39:44.492 --> 00:39:45.400 What is that? 00:39:50.874 --> 00:39:51.870 Sorry. 00:39:51.870 --> 00:39:54.900 Really threw me there. 00:39:54.900 --> 00:39:57.400 So probability of at least one head, 00:39:57.400 --> 00:40:02.685 it's not equal to the probability of head 00:40:02.685 --> 00:40:07.656 for the penny plus the probability of head 00:40:07.656 --> 00:40:13.130 for the dime, because we know this is really 3/4, 00:40:13.130 --> 00:40:15.220 and this is 1/2, this is 1/2. 00:40:15.220 --> 00:40:19.190 You add them up, 1/2 plus 1/2 does not equal to 3/4. 00:40:19.190 --> 00:40:22.890 So be careful. 00:40:22.890 --> 00:40:26.000 There's a lot of things you can say quickly that are not true. 00:40:28.940 --> 00:40:33.230 So anyway, you really have to consider the whole thing. 00:40:33.230 --> 00:40:36.304 And in the best case, if you can enumerate 00:40:36.304 --> 00:40:38.720 what's going to happen, it's very simple to add up things. 00:40:38.720 --> 00:40:41.380 Otherwise, you have to be very careful with the algebra 00:40:41.380 --> 00:40:43.506 to make sure you add all things correctly. 00:40:52.840 --> 00:40:55.000 Let's see at least what this should be equal to. 00:40:55.000 --> 00:40:56.890 So this should be equal to the probability 00:40:56.890 --> 00:41:03.900 that a head on a penny times the probability 00:41:03.900 --> 00:41:18.570 that I have head for the dime if I had a head for the penny 00:41:18.570 --> 00:41:23.696 plus probability of the tail for the dime 00:41:23.696 --> 00:41:30.244 or for the penny plus the probability that I 00:41:30.244 --> 00:41:39.450 have for the dime and then times something like this, too. 00:41:44.050 --> 00:41:46.810 And in a case like this, where I'm summing 00:41:46.810 --> 00:41:48.070 over all of the possibilities. 00:41:48.070 --> 00:41:49.540 So in this case, I'm either going 00:41:49.540 --> 00:41:50.740 to get a head for the dime or a tail for the dime. 00:41:50.740 --> 00:41:52.281 It's not going to balance on its end. 00:41:52.281 --> 00:41:54.760 I'm assuming that that chance is 0. 00:41:54.760 --> 00:42:00.636 Then these two things just add up to 1. 00:42:00.636 --> 00:42:03.044 Is that all right? 00:42:03.044 --> 00:42:04.960 It's really saying the probability of the dime 00:42:04.960 --> 00:42:09.590 will do something if I have a head for the penny. 00:42:09.590 --> 00:42:11.580 Is this all right? 00:42:11.580 --> 00:42:14.610 So you want to practice doing little algebra things like this 00:42:14.610 --> 00:42:16.068 to make sure you know how to do it. 00:42:20.500 --> 00:42:21.000 Yes? 00:42:21.000 --> 00:42:21.200 No? 00:42:21.200 --> 00:42:21.700 Maybe? 00:42:28.240 --> 00:42:30.518 Let's see what else I've got here. 00:42:49.748 --> 00:42:50.724 AUDIENCE: Professor. 00:42:50.724 --> 00:42:51.640 WILLIAM GREEN, JR: Yes 00:42:51.640 --> 00:42:54.340 AUDIENCE: What you just wrote, [INAUDIBLE] 00:42:54.340 --> 00:42:56.381 WILLIAM GREEN, JR: Yeah, what's the correct thing 00:42:56.381 --> 00:42:57.232 to write in here? 00:43:06.940 --> 00:43:07.940 What is the right thing? 00:43:12.833 --> 00:43:13.499 AUDIENCE: Tails. 00:43:32.300 --> 00:43:33.300 WILLIAM GREEN, JR: Yeah. 00:43:33.300 --> 00:43:34.130 So it's tricky. 00:43:34.130 --> 00:43:34.630 Yeah. 00:43:34.630 --> 00:43:40.521 AUDIENCE: The probability [INAUDIBLE] 00:43:40.521 --> 00:43:42.270 WILLIAM GREEN, JR: Yeah, it's the easy way 00:43:42.270 --> 00:43:44.320 to do it, for this case. 00:43:44.320 --> 00:43:46.367 And so there's, like, a lot of-- 00:43:46.367 --> 00:43:48.700 for some cases, it might be easier to write it this way. 00:43:48.700 --> 00:43:50.074 In some cases, write it that way. 00:43:50.074 --> 00:43:54.060 It depends on how many different options there are. 00:43:54.060 --> 00:43:58.665 So anyway, I'm just trying to warn you by writing this out. 00:43:58.665 --> 00:44:00.540 It's like, you might write down stuff quickly 00:44:00.540 --> 00:44:01.920 without thinking about it. 00:44:01.920 --> 00:44:03.360 And it's easy to double count. 00:44:03.360 --> 00:44:05.850 Like for example, if you put the other term in here, 00:44:05.850 --> 00:44:06.850 you double count. 00:44:06.850 --> 00:44:10.890 Because you already have, they had head, 00:44:10.890 --> 00:44:12.640 you already had the head case here. 00:44:21.830 --> 00:44:25.515 Yeah, maybe-- Did I write the-- 00:44:25.515 --> 00:44:27.890 I'll just write the base theorem down the way you usually 00:44:27.890 --> 00:44:29.370 see it. 00:44:29.370 --> 00:44:31.520 So this is the general expression, 00:44:31.520 --> 00:44:33.470 which is always true. 00:44:33.470 --> 00:44:43.970 The base theorem way to write it is the probability of A given B 00:44:43.970 --> 00:44:53.714 is equal to the probability of B given A times the probability 00:44:53.714 --> 00:44:59.490 of A over the probability of B. 00:44:59.490 --> 00:45:06.724 And that's just rearranging this equality. 00:45:06.724 --> 00:45:07.390 Rearrange again. 00:45:09.970 --> 00:45:14.007 And we'll come back to this with the situation 00:45:14.007 --> 00:45:16.090 that's like, what's the probability that Professor 00:45:16.090 --> 00:45:20.440 Green is 5' 9", given our measurements, 00:45:20.440 --> 00:45:23.650 is related to the probability that if Professor Green was 5' 00:45:23.650 --> 00:45:27.300 9", we would have made the measurements we got. 00:45:27.300 --> 00:45:30.510 So this is like that way to invert that statement. 00:45:39.780 --> 00:45:47.355 And if the thing that's here is exclusive, 00:45:47.355 --> 00:45:49.690 it means there's, like, many possible things that 00:45:49.690 --> 00:45:50.934 could happen. 00:45:50.934 --> 00:45:53.240 This is the probability that one of them happened. 00:45:53.240 --> 00:45:54.660 There's a lot of other things that could have happened. 00:45:54.660 --> 00:45:56.090 For instance, like heads and tails, it's either heads 00:45:56.090 --> 00:45:56.700 or it's tails. 00:45:56.700 --> 00:45:57.610 It's exclusive. 00:45:57.610 --> 00:46:06.730 If it's like that, you could rewrite that probability of B 00:46:06.730 --> 00:46:19.150 is equal to probability of Aj, probability of B given Aj 00:46:19.150 --> 00:46:21.940 summed over j. 00:46:21.940 --> 00:46:25.210 This is something, whatever A measurement 00:46:25.210 --> 00:46:29.700 it is, if it ends up with B, that's probability B. 00:46:29.700 --> 00:46:31.910 Is that all right? 00:46:31.910 --> 00:46:34.650 So you can put that into the denominator here. 00:46:34.650 --> 00:46:36.440 You can substitute that in so you 00:46:36.440 --> 00:46:41.633 can rewrite this as probability of Ai 00:46:41.633 --> 00:46:50.142 given B is equal to the probability of B given Ai 00:46:50.142 --> 00:46:54.600 times the probability of Ai divided 00:46:54.600 --> 00:47:06.722 by the sum of the probability of Aj [INAUDIBLE] Aj. 00:47:11.710 --> 00:47:14.112 And this is the form that you'll normally see based here. 00:47:20.870 --> 00:47:23.370 A lot of times, we have a continuous variable instead 00:47:23.370 --> 00:47:25.910 of discrete events like this. 00:47:25.910 --> 00:47:28.320 And so then we talk about probability distributions. 00:47:31.240 --> 00:47:33.980 And so instead of having a sum there, 00:47:33.980 --> 00:47:36.310 we might have an integral. 00:47:36.310 --> 00:47:42.400 And this is a topic that also is quite confusing to many people. 00:47:46.060 --> 00:47:51.956 So suppose I had a Maxwell-Boltzmann distribution. 00:47:51.956 --> 00:47:56.710 And I have, like, the probability density 00:47:56.710 --> 00:48:00.420 of having a certain velocity in the extraction instead 00:48:00.420 --> 00:48:02.440 of the particle. 00:48:02.440 --> 00:48:05.760 And so that's going to be something like e 00:48:05.760 --> 00:48:12.834 to the negative 1/2 mvx squared over keT divided by something. 00:48:15.663 --> 00:48:17.246 And maybe it actually is [INAUDIBLE].. 00:48:17.246 --> 00:48:18.080 I'm not sure. 00:48:18.080 --> 00:48:19.730 Maybe there's a vx here. 00:48:19.730 --> 00:48:20.900 Something like that. 00:48:20.900 --> 00:48:22.435 So you have an expression that you 00:48:22.435 --> 00:48:24.470 get for [INAUDIBLE] for probability density. 00:48:24.470 --> 00:48:25.970 Now, what does this mean? 00:48:25.970 --> 00:48:28.550 We want this thing to be the integral 00:48:28.550 --> 00:48:34.377 of P of vx dvx over from negative infinity to infinity. 00:48:34.377 --> 00:48:35.710 We want this to equal something. 00:48:35.710 --> 00:48:37.910 What do we want this to equal? 00:48:37.910 --> 00:48:39.120 1. 00:48:39.120 --> 00:48:42.176 So that means that the units of this, this 00:48:42.176 --> 00:48:44.709 is units of centimeters per second, 00:48:44.709 --> 00:48:45.750 what's the units of this? 00:48:45.750 --> 00:48:49.630 AUDIENCE: [INAUDIBLE] 00:48:51.120 --> 00:48:53.370 WILLIAM GREEN, JR: Centimeters per second now minus 1. 00:48:53.370 --> 00:48:57.860 It's, like, it's per the unit to this 00:48:57.860 --> 00:49:00.190 to get a dimensionless number there. 00:49:00.190 --> 00:49:05.506 So this has units of seconds per centimeter, 00:49:05.506 --> 00:49:07.880 which probably none of you thought until I just said that 00:49:07.880 --> 00:49:09.550 to you. 00:49:09.550 --> 00:49:14.030 So probability densities are tricky, 00:49:14.030 --> 00:49:17.679 and they always have to be multiplied by a delta. 00:49:17.679 --> 00:49:19.220 When I talk about this, I really need 00:49:19.220 --> 00:49:23.190 to talk about P of vx, delta vx. 00:49:23.190 --> 00:49:25.670 I need to have something here to make this look 00:49:25.670 --> 00:49:27.470 like a probability again. 00:49:27.470 --> 00:49:29.390 And so the issue is that the probability 00:49:29.390 --> 00:49:32.354 that the velocity is exactly something is, like, 0. 00:49:32.354 --> 00:49:33.770 It's really the probability that's 00:49:33.770 --> 00:49:37.630 in a certain range, plus or minus something. 00:49:37.630 --> 00:49:39.145 Then you get a nonzero probability. 00:49:43.372 --> 00:49:44.830 There's another quantity you'll see 00:49:44.830 --> 00:49:49.030 a lot called the cumulative probability distribution. 00:49:49.030 --> 00:49:50.460 Let's see, what letter do I use? 00:49:53.880 --> 00:49:56.650 Call it F. And this would be like the integral 00:49:56.650 --> 00:50:03.634 from negative infinity to vx prime, or vx of P over vx 00:50:03.634 --> 00:50:07.330 prime, dvx prime. 00:50:07.330 --> 00:50:09.880 And this is the probability that the particle 00:50:09.880 --> 00:50:11.830 has vx less than something. 00:50:11.830 --> 00:50:19.492 So this is F is equal to the probability that vx-- 00:50:19.492 --> 00:50:21.620 let's call this vx star-- 00:50:21.620 --> 00:50:26.520 vx is less than or equal to vx star. 00:50:26.520 --> 00:50:28.520 And that can be quite an important property. 00:50:28.520 --> 00:50:32.236 So for example, you're designing a supersonic nozzle. 00:50:32.236 --> 00:50:34.110 You want to know what gas molecules are going 00:50:34.110 --> 00:50:36.152 to come out at a certain speed. 00:50:36.152 --> 00:50:37.860 You really need to know that probability. 00:50:37.860 --> 00:50:40.565 How many of them are going to be bigger than that speed, how 00:50:40.565 --> 00:50:43.320 many less than that speed? 00:50:43.320 --> 00:50:47.562 So these are two different ways to express a similar thing. 00:50:47.562 --> 00:50:49.020 This is, like, the probably that it 00:50:49.020 --> 00:50:51.550 does have that speed within a certain range. 00:50:51.550 --> 00:50:54.240 This is the probability that it has anything less than 00:50:54.240 --> 00:50:55.680 or equal to that speed. 00:50:55.680 --> 00:50:58.020 And in a completely different way, this is an integral. 00:50:58.020 --> 00:51:02.640 This has units of dimensionless. 00:51:02.640 --> 00:51:06.400 This has units of per velocity. 00:51:06.400 --> 00:51:07.950 All right? 00:51:07.950 --> 00:51:10.860 All right, we'll pick up more of this on Monday.