WEBVTT 00:00:01.540 --> 00:00:03.910 The following content is provided under a Creative 00:00:03.910 --> 00:00:05.300 Commons license. 00:00:05.300 --> 00:00:07.510 Your support will help MIT OpenCourseWare 00:00:07.510 --> 00:00:11.600 continue to offer high quality educational resources for free. 00:00:11.600 --> 00:00:14.140 To make a donation or to view additional materials 00:00:14.140 --> 00:00:18.100 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.100 --> 00:00:25.260 at ocw.mit.edu 00:00:25.260 --> 00:00:28.880 WILLIAM GREEN: All right, so today we're 00:00:28.880 --> 00:00:30.860 going to keep on working on stochastic. 00:00:30.860 --> 00:00:32.280 And I guess maybe we start-- 00:00:32.280 --> 00:00:34.840 if you guys have questions from the homework set, 00:00:34.840 --> 00:00:38.010 since I'm sure you're thinking about that a lot right now. 00:00:38.010 --> 00:00:39.284 Are there any questions? 00:00:43.751 --> 00:00:46.000 Should I take this to mean that you have it completely 00:00:46.000 --> 00:00:46.583 under control? 00:00:49.281 --> 00:00:49.780 That's good. 00:00:52.370 --> 00:00:57.330 All right-- so since we have Monte Carlo integration 00:00:57.330 --> 00:01:00.700 metropolis totally under control, 00:01:00.700 --> 00:01:06.140 let's talk about time dependent probability distributions. 00:01:06.140 --> 00:01:16.420 So in many situations, we'd like to predict 00:01:16.420 --> 00:01:18.520 what's going to happen in an experiment. 00:01:18.520 --> 00:01:21.210 And the fundamental equations of that experiment 00:01:21.210 --> 00:01:23.590 are time dependent, so we have different things happening 00:01:23.590 --> 00:01:24.890 at different times. 00:01:24.890 --> 00:01:26.980 And so we really want to know what 00:01:26.980 --> 00:01:33.178 d dt of the probability of some observable-- 00:01:37.695 --> 00:01:39.070 this is going to equal something. 00:01:39.070 --> 00:01:39.986 Let me fix the lights. 00:01:47.810 --> 00:01:51.140 All right, so we want some equation like this. 00:01:51.140 --> 00:01:55.430 And then when you actually made your measurement, 00:01:55.430 --> 00:01:57.622 you'd be sampling from P. 00:01:57.622 --> 00:01:59.613 So probability of the observables-- 00:02:02.930 --> 00:02:06.250 and so if you made the measurement, 00:02:06.250 --> 00:02:08.750 you'd have a certain probability that you observe something. 00:02:08.750 --> 00:02:10.291 And you made a different measurement, 00:02:10.291 --> 00:02:12.680 you'd have same probability if you made the measurement 00:02:12.680 --> 00:02:13.360 at this time. 00:02:13.360 --> 00:02:17.270 So this is going to be probability of the observable 00:02:17.270 --> 00:02:18.580 depending on the time. 00:02:18.580 --> 00:02:20.790 And you made the measurements at different times. 00:02:20.790 --> 00:02:22.414 So let's conceptually think about this. 00:02:22.414 --> 00:02:24.560 Suppose we had a box. 00:02:24.560 --> 00:02:28.130 And it had a uranium atom in it that's radioactive. 00:02:28.130 --> 00:02:30.390 And it can decay. 00:02:30.390 --> 00:02:31.640 So we look in the box. 00:02:31.640 --> 00:02:33.650 And at time zero, I look in there. 00:02:33.650 --> 00:02:36.267 And there's the uranium atom sitting there nicely. 00:02:36.267 --> 00:02:37.350 And I go away for a while. 00:02:37.350 --> 00:02:37.850 They come back. 00:02:37.850 --> 00:02:38.641 And I open the box. 00:02:38.641 --> 00:02:39.742 And I look again later. 00:02:39.742 --> 00:02:42.200 And there's some probability that the uranium atom is still 00:02:42.200 --> 00:02:42.860 there. 00:02:42.860 --> 00:02:44.360 And there's some probability that it 00:02:44.360 --> 00:02:47.930 fissioned during the time and it's gone, right? 00:02:47.930 --> 00:02:50.360 OK so there must be some kind of equation 00:02:50.360 --> 00:02:53.390 like this, how the probability that the uranium atom 00:02:53.390 --> 00:02:56.150 is still there in the box. 00:02:56.150 --> 00:02:57.550 Right, is this OK-- 00:02:57.550 --> 00:02:58.050 yeah? 00:03:01.130 --> 00:03:03.725 So just for that case, what do we think the probability-- 00:03:03.725 --> 00:03:05.902 how does that probably behave-- 00:03:05.902 --> 00:03:06.520 any ideas? 00:03:13.100 --> 00:03:15.820 So uranium has a half-life or something, right? 00:03:15.820 --> 00:03:18.080 You guys heard this before, right? 00:03:18.080 --> 00:03:21.149 So let's just try to think of how this should scale. 00:03:21.149 --> 00:03:23.190 Whatever it is, it probably should have something 00:03:23.190 --> 00:03:28.320 to do with a time constant for how fast uranium decays, yeah? 00:03:31.815 --> 00:03:33.690 You guys are looking at me, like, so blankly. 00:03:33.690 --> 00:03:34.255 This is not that hard. 00:03:34.255 --> 00:03:35.379 It's just one uranium atom. 00:03:35.379 --> 00:03:37.150 It's OK. 00:03:37.150 --> 00:03:40.270 All right, so let's just write a plausible equation. 00:03:40.270 --> 00:03:43.780 Maybe it's dP dt, where this is the probability 00:03:43.780 --> 00:03:46.130 that the uranium is in the box. 00:03:50.020 --> 00:03:51.983 This might be the probability divided by two. 00:03:51.983 --> 00:03:54.130 Something like that-- would that be-- 00:03:54.130 --> 00:03:57.440 all right, I guess we need a negative sign maybe. 00:03:57.440 --> 00:03:59.890 So then the probability-- the solution to this 00:03:59.890 --> 00:04:03.850 would be that the probability that uranium is still there 00:04:03.850 --> 00:04:07.106 is going to be equal to some initial probability e 00:04:07.106 --> 00:04:10.816 to the negative time over tau. 00:04:10.816 --> 00:04:12.229 Does that look right? 00:04:15.060 --> 00:04:16.899 Does this look very reasonable? 00:04:16.899 --> 00:04:19.542 You might have seen this before, yeah? 00:04:19.542 --> 00:04:21.250 Maybe you didn't think of it probability. 00:04:21.250 --> 00:04:23.890 Maybe you thought about it as number of uraniums in the box. 00:04:23.890 --> 00:04:27.620 If I had written it as the number of uraniums 00:04:27.620 --> 00:04:30.890 is equal to the original number of uraniums e 00:04:30.890 --> 00:04:32.699 to the negative t over tau, you guys 00:04:32.699 --> 00:04:34.490 would have believed that right away, right? 00:04:38.400 --> 00:04:42.470 OK, so I guess maybe this is highlighting a issue here, 00:04:42.470 --> 00:04:44.810 is that we use a macroscopic thinking. 00:04:44.810 --> 00:04:48.099 We're used to thinking of things as continuum, right? 00:04:48.099 --> 00:04:50.390 But if I only have one uranium there, it's either there 00:04:50.390 --> 00:04:51.290 or it's not there. 00:04:51.290 --> 00:04:54.010 It's 1 or 0. 00:04:54.010 --> 00:04:56.740 So the problem with this equation-- 00:04:56.740 --> 00:05:02.660 if you compute this, it computes irrational numbers, 00:05:02.660 --> 00:05:05.087 fractional numbers as the number of uraniums. 00:05:05.087 --> 00:05:06.670 But of course, the number of uraniums, 00:05:06.670 --> 00:05:08.336 it's just either-- it's integers, right? 00:05:08.336 --> 00:05:11.320 There's one uranium atom, or there's two, or there's three. 00:05:11.320 --> 00:05:13.290 So this equation is not right, right? 00:05:13.290 --> 00:05:15.394 So this is incorrect. 00:05:15.394 --> 00:05:17.560 So that was what you learned in high school, right-- 00:05:17.560 --> 00:05:18.890 that equation? 00:05:18.890 --> 00:05:20.410 But that's not right. 00:05:20.410 --> 00:05:23.800 So this is really the correct equation, that the probability, 00:05:23.800 --> 00:05:26.350 that you have some number of uraniums that does something. 00:05:26.350 --> 00:05:28.410 You know-- well this works for one atom, anyway. 00:05:31.780 --> 00:05:34.120 So that's the equation for one. 00:05:34.120 --> 00:05:36.852 So this is probability I have one uranium. 00:05:36.852 --> 00:05:38.560 It's the initial probability that there's 00:05:38.560 --> 00:05:40.310 one uranium in a box, which is basically-- 00:05:40.310 --> 00:05:41.365 my case would be one. 00:05:41.365 --> 00:05:42.580 I know I put one in there. 00:05:45.250 --> 00:05:46.720 And then it just decays. 00:05:46.720 --> 00:05:48.862 The probability decays. 00:05:48.862 --> 00:05:50.320 But it doesn't mean uranium decays. 00:05:50.320 --> 00:05:53.095 So it means when I do-- if I do the experiment once, 00:05:53.095 --> 00:05:58.940 and here I have the uranium-- 00:05:58.940 --> 00:06:02.620 the probability of the uranium is a simple exponential decay-- 00:06:02.620 --> 00:06:04.420 time. 00:06:04.420 --> 00:06:05.830 I know I had a uranium atom there 00:06:05.830 --> 00:06:07.360 when I first put it there. 00:06:07.360 --> 00:06:09.826 And at later times it's gone away, 00:06:09.826 --> 00:06:11.492 right? 'Cause at longer and longer times 00:06:11.492 --> 00:06:13.370 it's more and more likely it would have fissioned. 00:06:13.370 --> 00:06:13.870 They 00:06:13.870 --> 00:06:15.640 Now if I actually do this. 00:06:15.640 --> 00:06:17.590 And make a-- buy a million boxes. 00:06:17.590 --> 00:06:19.265 And put a million uranium atoms-- one 00:06:19.265 --> 00:06:21.549 in each one to start with. 00:06:21.549 --> 00:06:23.090 And then I just keep coming back once 00:06:23.090 --> 00:06:26.320 in a while and checking, what I'll see, 00:06:26.320 --> 00:06:29.520 is that initially I had-- 00:06:29.520 --> 00:06:32.490 here's the number of uraniums. 00:06:32.490 --> 00:06:36.550 Initially I had a million uranium atoms. 00:06:36.550 --> 00:06:39.419 And for some time period I still had a million. 00:06:39.419 --> 00:06:41.210 And then something happened and one of them 00:06:41.210 --> 00:06:45.360 went away, because one of the fissioned, turned into lead-- 00:06:45.360 --> 00:06:47.570 or whatever they turn into. 00:06:47.570 --> 00:06:52.240 All right, and so now I had a million minus one uraniums. 00:06:52.240 --> 00:06:53.690 They live for a while. 00:06:53.690 --> 00:06:56.380 And then maybe another one went away. 00:06:56.380 --> 00:06:58.100 And then this time, another went away. 00:06:58.100 --> 00:07:00.830 And that time it lasted longer-- 00:07:00.830 --> 00:07:01.610 right? 00:07:01.610 --> 00:07:04.080 This is what you really expect to observe. 00:07:04.080 --> 00:07:06.350 Is this OK? 00:07:06.350 --> 00:07:13.536 And so this is sampling from the probability distribution 00:07:13.536 --> 00:07:14.400 that you would use. 00:07:14.400 --> 00:07:16.500 Does that make sense? 00:07:16.500 --> 00:07:19.620 So I'm sampling a million times, so I have a million boxes. 00:07:19.620 --> 00:07:22.550 This is the probability distribution for one box. 00:07:22.550 --> 00:07:25.550 And so I'm able to figure this out. 00:07:25.550 --> 00:07:26.300 Is this OK? 00:07:30.240 --> 00:07:34.050 All right, so we'd like to write equations like this 00:07:34.050 --> 00:07:37.187 for more complicated systems. 00:07:37.187 --> 00:07:39.270 But you'll still going to have the same confusion, 00:07:39.270 --> 00:07:41.869 that the probability is going to be continuous variable. 00:07:41.869 --> 00:07:43.410 And it's going to behave in ways that 00:07:43.410 --> 00:07:45.300 seem perfectly sensible to you. 00:07:45.300 --> 00:07:47.479 But then every time you do the experiment, 00:07:47.479 --> 00:07:49.020 different things will happen in time. 00:07:49.020 --> 00:07:51.660 Because you have something at the time-varying probability, 00:07:51.660 --> 00:07:53.700 and each time you sample from the probability distribution-- 00:07:53.700 --> 00:07:55.074 every time you do the experiment, 00:07:55.074 --> 00:07:57.780 it's like a particular instance of sampling 00:07:57.780 --> 00:07:59.250 from that probability distribution. 00:07:59.250 --> 00:08:03.705 And you might get that there is 87 uranium atoms left. 00:08:03.705 --> 00:08:06.470 And you might get that there's 93. 00:08:06.470 --> 00:08:09.030 And it's not because of your measurement error. 00:08:09.030 --> 00:08:13.320 It's because the intrinsic system has its own randomness 00:08:13.320 --> 00:08:14.304 to it, right? 00:08:14.304 --> 00:08:15.970 You know, you look at one uranium atom-- 00:08:15.970 --> 00:08:18.020 some uranium atoms live for a million years. 00:08:18.020 --> 00:08:21.130 And some uranium atoms might decay in the next second, 00:08:21.130 --> 00:08:21.630 right? 00:08:21.630 --> 00:08:23.780 And you have no idea when they're going to decay. 00:08:23.780 --> 00:08:27.560 All you know is statistically, on the average, uraniums decay 00:08:27.560 --> 00:08:30.410 with a certain half-life, which is pretty long, right? 00:08:33.692 --> 00:08:35.150 All right, is this totally obvious? 00:08:35.150 --> 00:08:36.066 I can't figure it out. 00:08:36.066 --> 00:08:38.559 Is this, like, totally obvious, or totally confusing? 00:08:38.559 --> 00:08:40.809 It's very funny-- with your eyes and your expressions, 00:08:40.809 --> 00:08:42.517 totally obvious and totally confusing has 00:08:42.517 --> 00:08:43.660 the same blank expression. 00:08:47.100 --> 00:08:48.290 Can someone ask a question. 00:08:48.290 --> 00:08:52.130 Maybe that would help us to have a-- 00:08:52.130 --> 00:08:52.967 this is OK? 00:08:52.967 --> 00:08:54.050 All right, this is the OK. 00:08:54.050 --> 00:08:54.400 AUDIENCE: It's OK. 00:08:54.400 --> 00:08:55.100 WILLIAM GREEN: This is OK. 00:08:55.100 --> 00:08:56.600 All right, so it's totally obvious-- 00:08:56.600 --> 00:08:58.700 good, hard to tell sometimes. 00:08:58.700 --> 00:09:03.080 All right, so Joe Scott wrote some brilliant notes trying 00:09:03.080 --> 00:09:05.122 to explain how do this for chemical kinetics, OK? 00:09:05.122 --> 00:09:06.579 So you should definitely read them. 00:09:06.579 --> 00:09:08.810 They're in the-- posted under the materials, 00:09:08.810 --> 00:09:10.530 general section-- 00:09:10.530 --> 00:09:13.610 Stochastic Chemical Kinetics-- nice 20 page long thing, 00:09:13.610 --> 00:09:14.810 something like that. 00:09:14.810 --> 00:09:16.690 It's definitely worthwhile to read it. 00:09:16.690 --> 00:09:19.670 He was very nice at writing very clear notes. 00:09:19.670 --> 00:09:23.180 So I'll try to explain his notes inexpertly, 00:09:23.180 --> 00:09:24.440 in a short period of time. 00:09:24.440 --> 00:09:26.160 I strongly recommend you read them. 00:09:26.160 --> 00:09:32.140 So the idea is that the chemical kinetic equations 00:09:32.140 --> 00:09:35.390 you guys have all used are also incorrect, 00:09:35.390 --> 00:09:36.950 right-- just like this is incorrect. 00:09:39.860 --> 00:09:42.260 Because this is assuming that something that's 00:09:42.260 --> 00:09:45.470 really a discrete variable is a continuous variable. 00:09:45.470 --> 00:09:47.450 And we do this all the time, right? 00:09:47.450 --> 00:09:50.470 We know that material is made of atoms. 00:09:50.470 --> 00:09:54.110 And so when you measure our mass in a certain volume, 00:09:54.110 --> 00:09:57.180 there only can be certain discrete values. 00:09:57.180 --> 00:09:58.470 But we always ignore that. 00:09:58.470 --> 00:10:01.020 And we always treat is a continuous variable, right? 00:10:01.020 --> 00:10:02.070 So you have some flow-- 00:10:02.070 --> 00:10:03.665 some amount of liquid-- 00:10:03.665 --> 00:10:05.040 and you have some amount of mass. 00:10:05.040 --> 00:10:06.570 And it could be any number. 00:10:06.570 --> 00:10:11.935 And we treat dm dt, or things like dm dx, d rho dx-- 00:10:11.935 --> 00:10:13.560 we write those down as if everything is 00:10:13.560 --> 00:10:15.400 perfectly continuous variables. 00:10:15.400 --> 00:10:18.000 And that's because that in a lot of systems we have, 00:10:18.000 --> 00:10:20.880 we have so many atoms-- 00:10:20.880 --> 00:10:22.830 and we can't measure that well anyway-- 00:10:22.830 --> 00:10:25.680 that we couldn't tell if we were missing one or two atoms. 00:10:25.680 --> 00:10:28.890 Whether we end up with an extra half an atom in our equations 00:10:28.890 --> 00:10:31.421 or not makes no difference, because we're not that precise, 00:10:31.421 --> 00:10:31.920 right? 00:10:31.920 --> 00:10:33.045 So we don't worry about it. 00:10:33.045 --> 00:10:35.720 So we use contiuum equations everywhere and-- 00:10:35.720 --> 00:10:38.930 but in many cases, what we really should be using 00:10:38.930 --> 00:10:42.490 are continuous equations for probabilities. 00:10:42.490 --> 00:10:45.070 And then as going to show you how you do it for chemical 00:10:45.070 --> 00:10:46.380 kinetics in a minute. 00:10:46.380 --> 00:10:48.460 I'd say that this is super fundamental, actually. 00:10:48.460 --> 00:10:51.850 So quantum mechanics says that everything 00:10:51.850 --> 00:10:54.370 is wave functions, which is actually 00:10:54.370 --> 00:10:57.260 like the square root of a probability density. 00:10:57.260 --> 00:11:02.150 And the real equations are really probability densities. 00:11:02.150 --> 00:11:04.940 And so that's the real fundamental equations. 00:11:04.940 --> 00:11:06.441 Really, everything is probabilities. 00:11:06.441 --> 00:11:08.065 And when you make an experiment, you're 00:11:08.065 --> 00:11:09.530 just sampling from the probability. 00:11:09.530 --> 00:11:12.800 And all our continuum equations are like approximations. 00:11:12.800 --> 00:11:19.150 They're only valid in, like, the large number of atoms limit, 00:11:19.150 --> 00:11:21.720 OK? 00:11:21.720 --> 00:11:24.055 I know this takes maybe a change of thought, 00:11:24.055 --> 00:11:26.180 because you're so used to those continuum equations 00:11:26.180 --> 00:11:29.370 you start to believe they're really true, right? 00:11:29.370 --> 00:11:31.010 You've probably used them a lot. 00:11:31.010 --> 00:11:33.176 And you start to feel like you love those equations. 00:11:33.176 --> 00:11:34.550 They're your favorite equations. 00:11:34.550 --> 00:11:36.170 You know, you have a relationship with them. 00:11:36.170 --> 00:11:37.880 Some of them you might hate, some you love. 00:11:37.880 --> 00:11:40.130 But at least, you have a strong connection with them. 00:11:40.130 --> 00:11:41.660 And you just find out that actually they're 00:11:41.660 --> 00:11:43.090 not even related to the real equations. 00:11:43.090 --> 00:11:45.256 The real equations are actually something different. 00:11:45.256 --> 00:11:47.924 This can be very annoying, but it's true. 00:11:47.924 --> 00:11:50.090 And when you get to smaller and smaller systems that 00:11:50.090 --> 00:11:52.801 have fewer and fewer molecules in them, 00:11:52.801 --> 00:11:54.800 then you get more and more sensitive to the fact 00:11:54.800 --> 00:11:57.240 that the regular equations are incorrect. 00:11:57.240 --> 00:12:00.880 And this comes up a lot in small systems. 00:12:00.880 --> 00:12:02.810 So many of you probably will do research 00:12:02.810 --> 00:12:05.560 that has to do with biology. 00:12:05.560 --> 00:12:07.040 In biology there's cells. 00:12:07.040 --> 00:12:09.830 Inside cells there's subcellular structures. 00:12:09.830 --> 00:12:11.720 Inside those subcellular structures, 00:12:11.720 --> 00:12:13.950 there's not very many molecules. 00:12:13.950 --> 00:12:17.660 And so you often are at the limit where you have, like, 00:12:17.660 --> 00:12:20.780 one molecule of a certain type inside this mitochondria, 00:12:20.780 --> 00:12:21.940 or something. 00:12:21.940 --> 00:12:23.390 And that's it. 00:12:23.390 --> 00:12:27.440 And so you wouldn't expect the continuum equations to work. 00:12:27.440 --> 00:12:29.330 And in fact there's a big field-- 00:12:29.330 --> 00:12:31.130 many people at research do in here-- 00:12:31.130 --> 00:12:33.230 called single molecule spectroscopy, 00:12:33.230 --> 00:12:36.040 where you can measure the motion of one DNA molecule 00:12:36.040 --> 00:12:37.115 or something like that. 00:12:37.115 --> 00:12:41.870 And so you really see the individual molecules, 00:12:41.870 --> 00:12:46.610 and the effects of the individuality of the molecules. 00:12:46.610 --> 00:12:48.995 And this is not only true in biology. 00:12:51.356 --> 00:12:52.730 For a long time-- there's a field 00:12:52.730 --> 00:12:55.040 called emulsion polymerization. 00:12:55.040 --> 00:12:56.380 You guys ever hear of this? 00:12:56.380 --> 00:12:57.838 So like, all the paint on the walls 00:12:57.838 --> 00:13:00.259 here is made by that process, where they 00:13:00.259 --> 00:13:01.550 polymerize methyl methacrylate. 00:13:01.550 --> 00:13:04.070 They want to make the little beads of polymers 00:13:04.070 --> 00:13:07.900 all have a small dispersion as possible, 00:13:07.900 --> 00:13:10.160 to make all the molecules the same. 00:13:10.160 --> 00:13:12.430 So what they do is they make a colloidal suspension-- 00:13:12.430 --> 00:13:14.050 I'm going to draw a little picture. 00:13:14.050 --> 00:13:15.890 So they have the reactor. 00:13:15.890 --> 00:13:20.570 They make little bubbles of the monomer that 00:13:20.570 --> 00:13:22.920 are suspended in some solid. 00:13:22.920 --> 00:13:24.840 And inside each of these bubbles-- 00:13:24.840 --> 00:13:26.465 because the way they make them, they're 00:13:26.465 --> 00:13:27.504 all about the same size. 00:13:27.504 --> 00:13:28.920 So they all have a lot of bubbles, 00:13:28.920 --> 00:13:30.000 all about the same size. 00:13:30.000 --> 00:13:31.565 And then they polymerize. 00:13:31.565 --> 00:13:32.940 And they polymerize each of them. 00:13:32.940 --> 00:13:33.870 And they polymerize. 00:13:33.870 --> 00:13:35.703 And basically, all the material that's there 00:13:35.703 --> 00:13:38.100 polymerizes into one molecule. 00:13:38.100 --> 00:13:40.440 Now how do they make that happen-- 00:13:40.440 --> 00:13:43.800 is that they introduce, say, a free radical. 00:13:43.800 --> 00:13:47.160 And one free radical gets into one of these bubbles. 00:13:47.160 --> 00:13:48.960 And it can polymerize the whole thing, 00:13:48.960 --> 00:13:50.700 because the reactions are a radical plus 00:13:50.700 --> 00:13:54.840 monomer goes to a polymer-- 00:13:54.840 --> 00:13:55.860 bigger polymer, right? 00:13:55.860 --> 00:14:01.610 So polymer radical size n goes to polymer 00:14:01.610 --> 00:14:03.400 radical size n plus 1. 00:14:03.400 --> 00:14:06.020 That's the polymerization reaction. 00:14:06.020 --> 00:14:08.910 Now normally, if you did this in the bulk, 00:14:08.910 --> 00:14:11.110 there would be a lot of radicals. 00:14:11.110 --> 00:14:16.420 And some of these guys would react with each other. 00:14:19.430 --> 00:14:21.930 And the radical will be gone, because the two radicals would 00:14:21.930 --> 00:14:25.720 react, make a new chemical bond, and this process would stop. 00:14:25.720 --> 00:14:29.110 And so this gives a big dispersion in the size 00:14:29.110 --> 00:14:30.950 of the polymer chains you get. 00:14:30.950 --> 00:14:33.550 So to beat this, they do emulsion polymerization. 00:14:33.550 --> 00:14:35.890 And they make that number of radicals so small 00:14:35.890 --> 00:14:38.740 that statistically, at any time, any of these droplets 00:14:38.740 --> 00:14:42.930 either has zero or one of radicals in it. 00:14:42.930 --> 00:14:45.470 And there's not-- during the time 00:14:45.470 --> 00:14:47.690 it does the polymerization, there's not enough time 00:14:47.690 --> 00:14:49.114 for a second radical to come in. 00:14:49.114 --> 00:14:51.155 So it's only got one radical, so that one radical 00:14:51.155 --> 00:14:52.779 is going to eat up every single monomer 00:14:52.779 --> 00:14:54.524 molecule in the whole thing. 00:14:54.524 --> 00:14:56.440 And so it's going take a giant polymer, that's 00:14:56.440 --> 00:14:59.370 a whole size of the emulsion. 00:14:59.370 --> 00:15:02.642 If you made the drops too big, you get two radicals in there. 00:15:02.642 --> 00:15:04.100 And you get some of this happening. 00:15:04.100 --> 00:15:05.760 And then you get a dispersion. 00:15:05.760 --> 00:15:09.670 So this is emulsion polymerization. 00:15:14.080 --> 00:15:18.320 All right, so cells have small little volumes. 00:15:18.320 --> 00:15:21.110 Colloids and emulsions have small little volumes. 00:15:21.110 --> 00:15:22.790 They all have-- that the-- 00:15:22.790 --> 00:15:26.697 some of the molecules-- the important molecules are so-- 00:15:26.697 --> 00:15:29.030 have such low concentrations-- the volumes are so small, 00:15:29.030 --> 00:15:31.850 that the concentration times the volume is less than 1, 00:15:31.850 --> 00:15:33.290 or about 1. 00:15:33.290 --> 00:15:37.370 So that's for like-- you know, your copies of DNA in you, 00:15:37.370 --> 00:15:38.550 in your cells. 00:15:38.550 --> 00:15:42.800 It's the same kind of thing for radicals-- free radicals 00:15:42.800 --> 00:15:44.570 in a polymerization system. 00:15:44.570 --> 00:15:47.194 And similar things could happen with, like, explosions-- 00:15:47.194 --> 00:15:48.860 different kinds of rare events, that you 00:15:48.860 --> 00:15:50.734 might have a very small number of some really 00:15:50.734 --> 00:15:52.010 important molecule. 00:15:52.010 --> 00:15:53.900 And then whether or not you have one or not 00:15:53.900 --> 00:15:57.390 is really important, OK. 00:15:57.390 --> 00:16:00.490 All right, so we want to model those kinds of systems, 00:16:00.490 --> 00:16:02.370 we have to worry about the graininess. 00:16:02.370 --> 00:16:04.665 So we want to write down what the equation is dP dt. 00:16:08.240 --> 00:16:12.050 So first you've got to think, well what is P? 00:16:12.050 --> 00:16:19.625 Let's do a case where I have the reaction of A plus B 00:16:19.625 --> 00:16:21.717 goes to C, OK? 00:16:21.717 --> 00:16:23.300 Suppose this is the only reaction that 00:16:23.300 --> 00:16:24.500 can happen in our system. 00:16:24.500 --> 00:16:26.333 The only molecules we have in our system are 00:16:26.333 --> 00:16:27.640 A's, B's, and C's. 00:16:27.640 --> 00:16:29.630 And so our probability distribution-- 00:16:29.630 --> 00:16:31.130 our probability that we want to know 00:16:31.130 --> 00:16:34.170 is how many A's, how many B's, how many C's do 00:16:34.170 --> 00:16:38.664 we have in our system, OK? 00:16:38.664 --> 00:16:41.940 So that's the probability we're interested to know. 00:16:41.940 --> 00:16:45.210 We could have 5 A's, 2 B's, and 0 C's. 00:16:45.210 --> 00:16:49.720 We could have 3 A's, 7 B's, 11 C's-- who knows, right-- 00:16:49.720 --> 00:16:51.020 different numbers like that. 00:16:51.020 --> 00:16:53.396 If it gets out to be 10 to the 23, then there's no point. 00:16:53.396 --> 00:16:55.353 You don't have to do this, because you can just 00:16:55.353 --> 00:16:57.420 go back your continuum equations, right? 00:16:57.420 --> 00:17:00.450 But if you have 5, or 3, or 2, or 1, or 0, 00:17:00.450 --> 00:17:03.390 then you really need to worry about the graininess. 00:17:03.390 --> 00:17:07.190 So you want to write down an equation. 00:17:07.190 --> 00:17:13.819 So here's a probability of n A n B n C. 00:17:13.819 --> 00:17:20.214 And it's going to equal to reactions which consume-- 00:17:20.214 --> 00:17:22.130 if I'm in this state that has a certain number 00:17:22.130 --> 00:17:25.670 A's, B's, and C's, and I can react away, 00:17:25.670 --> 00:17:27.440 so I only have reactions-- 00:17:27.440 --> 00:17:29.090 for example, I'm in this state. 00:17:29.090 --> 00:17:32.020 An d plus B could react with each other, 00:17:32.020 --> 00:17:33.620 and will react away. 00:17:33.620 --> 00:17:35.316 So I'll have some reaction. 00:17:35.316 --> 00:17:36.690 I'm going to write a k, but we'll 00:17:36.690 --> 00:17:39.780 come back to what this really means exactly in a minute. 00:17:39.780 --> 00:17:46.040 So this is the k for A plus B going to C. 00:17:46.040 --> 00:17:47.840 And we think that this is going to scale 00:17:47.840 --> 00:17:53.370 with the number of A's and the number of B's I've got, right? 00:17:53.370 --> 00:18:01.670 And this depends on the probability that I have A B C. 00:18:01.670 --> 00:18:08.310 So if I'm in this state with n A's n B's n C's-- 00:18:08.310 --> 00:18:09.970 that probability is going to go down, 00:18:09.970 --> 00:18:14.330 because the A's and B's can react with each other, right? 00:18:14.330 --> 00:18:18.900 Now on the other hand, I have plus the same k. 00:18:22.790 --> 00:18:26.460 And this is going to be n A prime n B prime. 00:18:35.890 --> 00:18:40.630 If I have some other state that can react so that it makes 00:18:40.630 --> 00:18:42.730 this state, then I'll get a plus sign, 00:18:42.730 --> 00:18:44.990 because probability of this state would increase. 00:18:44.990 --> 00:18:52.700 So for example, if n A prime is equal to n A plus 1. 00:18:52.700 --> 00:18:57.450 And n B prime is equal to n B plus 1 00:18:57.450 --> 00:19:02.720 and n C prime is equal to n C minus 1, 00:19:02.720 --> 00:19:06.370 then that state can react by one of these A's and one 00:19:06.370 --> 00:19:09.220 of these B's combining together to make a C. That will get me 00:19:09.220 --> 00:19:11.850 to this exact state here. 00:19:11.850 --> 00:19:14.710 Does this make sense? 00:19:14.710 --> 00:19:19.294 All Right, so that's the master equation 00:19:19.294 --> 00:19:20.710 if I have a system that could only 00:19:20.710 --> 00:19:25.860 do the irreversible reaction A plus B goes to C. So the only-- 00:19:25.860 --> 00:19:28.120 I can go away-- because the probability density 00:19:28.120 --> 00:19:30.520 of this-- the probability that this state can reduce, 00:19:30.520 --> 00:19:33.310 because that reaction occurs starting from this state. 00:19:33.310 --> 00:19:35.620 And probability this state can increase, 00:19:35.620 --> 00:19:37.680 because I started from this other state here-- 00:19:37.680 --> 00:19:39.040 this special state here. 00:19:39.040 --> 00:19:42.230 And that would make the state interest. 00:19:42.230 --> 00:19:45.020 All right and so I have equation-- this equation 00:19:45.020 --> 00:19:47.750 over and over again, for every possible value of n A, n B, 00:19:47.750 --> 00:19:50.140 n C. Is this OK? 00:19:52.890 --> 00:19:55.200 Now I have to actually-- 00:19:55.200 --> 00:19:58.000 remember for chemical kinetics, that whenever 00:19:58.000 --> 00:20:01.990 you have a forward reaction, you also have the reverse reaction. 00:20:01.990 --> 00:20:05.800 So there's actually a couple more terms. 00:20:05.800 --> 00:20:08.010 So now I have another loss term. 00:20:08.010 --> 00:20:12.180 This is C goes to A plus B. This is going to tell of the number 00:20:12.180 --> 00:20:13.920 of C's. 00:20:13.920 --> 00:20:20.100 And that's P of n A n B n C. And then 00:20:20.100 --> 00:20:21.470 I have another source term. 00:20:28.210 --> 00:20:31.130 And this is where I start from the state with one more 00:20:31.130 --> 00:20:50.360 C. Is that OK? 00:20:50.360 --> 00:20:55.370 So this set of equations is called the kinetic master 00:20:55.370 --> 00:20:57.410 equation. 00:20:57.410 --> 00:21:00.750 And this is the equation that describes the real system. 00:21:00.750 --> 00:21:04.490 So this is the real probability of finding the system with any 00:21:04.490 --> 00:21:07.350 number of A's, B's, and C's. 00:21:07.350 --> 00:21:10.560 And I have an equation like this for every different possible 00:21:10.560 --> 00:21:14.630 number of A's, number of B's, number of C's. 00:21:14.630 --> 00:21:17.360 All right? 00:21:17.360 --> 00:21:20.350 And it's not bad as an equation goes. 00:21:20.350 --> 00:21:22.840 It's a linear differential equation, 00:21:22.840 --> 00:21:25.090 because it's linear in P's, right? 00:21:25.090 --> 00:21:29.140 There's just first-- everything's first power in P. 00:21:29.140 --> 00:21:38.950 So in general, I can rewrite this as dP dt is the some 00:21:38.950 --> 00:21:45.730 matrix times B. And the matrix entries are these prefactors-- 00:21:45.730 --> 00:21:48.910 things like this and this-- 00:21:48.910 --> 00:21:52.550 sorry, this, this. 00:21:52.550 --> 00:21:54.750 These are the different elements in the matrix, m. 00:22:00.790 --> 00:22:06.790 So you should go ahead and be able to write 00:22:06.790 --> 00:22:09.900 any kinetic equation system-- rewrite it in this form 00:22:09.900 --> 00:22:11.040 if you want. 00:22:11.040 --> 00:22:12.740 It's called the kinetic master equation. 00:22:23.751 --> 00:22:25.560 Now if I do an experiment like I talked 00:22:25.560 --> 00:22:28.980 about with uranium, where I put one uranium atom in there. 00:22:28.980 --> 00:22:30.700 And all uranium atoms can react to do 00:22:30.700 --> 00:22:32.650 is fall apart to lead plus neutrons, 00:22:32.650 --> 00:22:34.382 or whatever the heck they fall apart to. 00:22:34.382 --> 00:22:36.340 So I can write down-- there's like one reaction 00:22:36.340 --> 00:22:38.470 that uranium nuclei do. 00:22:38.470 --> 00:22:39.630 And so I can write it out. 00:22:39.630 --> 00:22:41.880 It will just be something like this. 00:22:41.880 --> 00:22:43.714 And this equation will work for a case 00:22:43.714 --> 00:22:45.880 where I put one uranium atom in there to begin with. 00:22:45.880 --> 00:22:48.296 I could do the case where I put two uranium atoms in there 00:22:48.296 --> 00:22:49.360 to begin with. 00:22:49.360 --> 00:22:50.605 I could write this down. 00:22:50.605 --> 00:22:52.720 Is that all right? 00:22:52.720 --> 00:23:02.230 OK, now if I had a case where I had 27 A's, 14 B's, and 33 C's 00:23:02.230 --> 00:23:05.230 to start with, then you can see that the number 00:23:05.230 --> 00:23:08.490 of possibilities here might get pretty large. 00:23:08.490 --> 00:23:11.140 'Cause I have to consider, well, I could lose one A, 00:23:11.140 --> 00:23:13.540 and lose one B, and get one more C. 00:23:13.540 --> 00:23:16.330 But then I'll have an equation for this term which 00:23:16.330 --> 00:23:17.510 will look just like this. 00:23:17.510 --> 00:23:19.880 But it'll go to ones where I have lost two A's and two 00:23:19.880 --> 00:23:21.485 B's and gained two C's. 00:23:21.485 --> 00:23:24.110 And then all the way down, until you get down to one of the n's 00:23:24.110 --> 00:23:26.630 to be zero. 00:23:26.630 --> 00:23:29.990 And so it could be quite a lot of possible equations, right? 00:23:29.990 --> 00:23:34.610 So the dimension-- the length of the vector P 00:23:34.610 --> 00:23:38.250 might get really long, right? 00:23:38.250 --> 00:23:52.110 So the length of P is something like the max number of A's, max 00:23:52.110 --> 00:23:56.060 number of B's, max number of C's. 00:24:00.690 --> 00:24:02.100 All right, so if I have-- 00:24:02.100 --> 00:24:06.010 I'll allow-- consider systems with up to 10 A's, 10 B's, 10 00:24:06.010 --> 00:24:07.140 C's-- 00:24:07.140 --> 00:24:09.660 then this is 1,000-- 00:24:09.660 --> 00:24:11.492 P's a length of approximately 1,000. 00:24:11.492 --> 00:24:13.700 And it might be less than that, because maybe there's 00:24:13.700 --> 00:24:17.160 some of them-- some particular states I can't get to. 00:24:17.160 --> 00:24:19.796 So maybe it's less than 1,000, but something like 1,000. 00:24:19.796 --> 00:24:21.670 On the other hand, if I allow it to a million 00:24:21.670 --> 00:24:24.390 A's, a million B's, and millions C's, then I, 00:24:24.390 --> 00:24:28.950 like, 10 to the 18 length of P. So P gets pretty long, pretty 00:24:28.950 --> 00:24:33.990 fast when the number of possible states goes up. 00:24:33.990 --> 00:24:35.990 Is that all right? 00:24:35.990 --> 00:24:40.780 So-- now, it's linear though. 00:24:40.780 --> 00:24:41.450 And it's linear. 00:24:41.450 --> 00:24:42.866 And this is constant coefficients. 00:24:42.866 --> 00:24:45.030 And so this is really not-- and sparse. 00:24:45.030 --> 00:24:47.460 So this is really-- it's not that bad. 00:24:47.460 --> 00:24:50.100 And in fact Professor Barton's group 00:24:50.100 --> 00:24:51.690 has worked on these kind of problems. 00:24:51.690 --> 00:24:54.450 And he has to solves that can work with P has dimension up 00:24:54.450 --> 00:24:56.410 to 200 million. 00:24:56.410 --> 00:24:58.960 So if you can keep the number-- say it's 200 million or less, 00:24:58.960 --> 00:25:01.360 you can actually solve this exactly. 00:25:01.360 --> 00:25:03.400 Now solving this exactly is really good, 00:25:03.400 --> 00:25:06.010 because then you actually have the exact probability 00:25:06.010 --> 00:25:09.620 of any observable measurement that you would possibly make 00:25:09.620 --> 00:25:13.390 at any time, all computed, OK? 00:25:13.390 --> 00:25:15.010 So that's really great. 00:25:15.010 --> 00:25:17.136 But as I just pointed out, if I make these numbers, 00:25:17.136 --> 00:25:19.176 like, 10 to the 6, 10 to the 6, and 10 to the 6-- 00:25:19.176 --> 00:25:20.380 this is like 10 of the 18. 00:25:20.380 --> 00:25:23.130 That's a lot bigger than 200 million. 00:25:23.130 --> 00:25:27.600 And so Professor Barton's numerical methods 00:25:27.600 --> 00:25:30.310 are going to poop out at some point, 00:25:30.310 --> 00:25:32.260 and not going to work for you anymore. 00:25:32.260 --> 00:25:34.770 So there's a limited number of problems you can do. 00:25:34.770 --> 00:25:38.010 So if your problem small enough, you can just solve this-- 00:25:38.010 --> 00:25:43.620 for really small ones you can solve this with OD45 or OD15s 00:25:43.620 --> 00:25:46.844 But because it's linear, there's special solution methods. 00:25:46.844 --> 00:25:49.260 I don't know if you remember solutions of linear equations 00:25:49.260 --> 00:25:52.110 this, you can relate them to the eigenvectors 00:25:52.110 --> 00:25:54.060 and eigenvalues of m. 00:25:54.060 --> 00:25:56.484 And so you can do-- that's one way to do it. 00:25:56.484 --> 00:25:58.400 And then there are special methods you can do. 00:25:58.400 --> 00:25:59.441 Talk to Professor Barton. 00:25:59.441 --> 00:26:02.421 If it's sparse-- and it has certain kinds properties, 00:26:02.421 --> 00:26:04.920 you can make this really-- you can make quite large systems, 00:26:04.920 --> 00:26:08.820 up to 10 to the 8 kind of size. 00:26:08.820 --> 00:26:11.470 If we get a system with more than 10 to 8 possible states, 00:26:11.470 --> 00:26:13.470 that means more than 10 to the 8 possible things 00:26:13.470 --> 00:26:16.770 could be observed or the result of experiment 00:26:16.770 --> 00:26:19.169 could have more than 10 to 8 possibilities, 00:26:19.169 --> 00:26:20.710 then this is-- you're not going to be 00:26:20.710 --> 00:26:21.970 able to solve it this way. 00:26:21.970 --> 00:26:23.553 And so then we're going to have to use 00:26:23.553 --> 00:26:26.390 a stochastic solid solution method to try to figure-- 00:26:26.390 --> 00:26:31.290 to do-- so we're going to try a sample from P of t. 00:26:31.290 --> 00:26:33.910 So I guess I should comment, again, I wrote it this way. 00:26:33.910 --> 00:26:34.729 But the thing-- 00:26:34.729 --> 00:26:36.270 P is definitely-- it depends on time. 00:26:36.270 --> 00:26:38.228 So it's you'll have different P for every time. 00:26:41.600 --> 00:26:43.315 Yeah? 00:26:43.315 --> 00:26:58.167 AUDIENCE: [INAUDIBLE] 00:26:58.167 --> 00:27:00.250 WILLIAM GREEN: Yeah maybe I'll just do better to-- 00:27:00.250 --> 00:27:02.250 I can substitute it in here. 00:27:17.250 --> 00:27:18.750 Is that all right? 00:27:18.750 --> 00:27:20.958 AUDIENCE: So, like, if you transfer all the neighbors 00:27:20.958 --> 00:27:22.750 in it, that's--? 00:27:22.750 --> 00:27:27.220 WILLIAM GREEN: Yeah, 'cause a key idea 00:27:27.220 --> 00:27:34.720 is that chemical kinetics only does one step at a time. 00:27:34.720 --> 00:27:41.954 So if you have a reaction like this, at any instant 00:27:41.954 --> 00:27:43.370 the probability that two reactions 00:27:43.370 --> 00:27:45.453 are going to be happening exactly the same instant 00:27:45.453 --> 00:27:46.770 is negligible. 00:27:46.770 --> 00:27:50.150 So at any instant you're only changing the numbers 00:27:50.150 --> 00:27:52.820 of the atoms by one unit, yeah. 00:27:52.820 --> 00:27:55.664 AUDIENCE: Then shouldn't you add another A minus 1 00:27:55.664 --> 00:27:56.612 and an E minus 1? 00:27:59.731 --> 00:28:01.480 WILLIAM GREEN: Maybe I got this backwards. 00:28:01.480 --> 00:28:07.680 n A plus 1 plus 1 minus 1, minus 1 minus 1 plus 1-- 00:28:07.680 --> 00:28:11.330 I think got-- I think this is right. 00:28:11.330 --> 00:28:14.600 So this is for the forward reaction that always reduces 00:28:14.600 --> 00:28:15.827 the number of A's and B's. 00:28:15.827 --> 00:28:18.326 And this is the reverse reaction that always increases them. 00:28:18.326 --> 00:28:20.117 And so I think you generally get like this. 00:28:20.117 --> 00:28:21.680 You always get-- for any reaction, 00:28:21.680 --> 00:28:24.190 you'll have four terms-- 00:28:24.190 --> 00:28:26.650 so the forward and reverse, starting 00:28:26.650 --> 00:28:29.385 from initial state, and the forward and reverse 00:28:29.385 --> 00:28:30.260 that make your state. 00:28:32.762 --> 00:28:34.220 And then if you add more reactions, 00:28:34.220 --> 00:28:35.540 you have a system where you've got a lot more reactions. 00:28:35.540 --> 00:28:38.150 Maybe you have C's that can just decompose 00:28:38.150 --> 00:28:40.580 into B's or something-- different reactions. 00:28:40.580 --> 00:28:44.150 Then you'd have a sum like this with terms 00:28:44.150 --> 00:28:45.950 like this from every reaction. 00:28:45.950 --> 00:28:48.283 And you'd go through all your list of all your reaction. 00:28:50.647 --> 00:28:52.730 I guess there's one funny thing I should point out 00:28:52.730 --> 00:28:56.120 to you about this, which is very important for this emulsion 00:28:56.120 --> 00:28:59.030 polymerization case-- 00:28:59.030 --> 00:29:07.480 Is that if you have a reaction of the type A plus A goes to-- 00:29:07.480 --> 00:29:14.900 I don't know-- B. That might be a reaction you could have. 00:29:14.900 --> 00:29:18.350 That when I write down the equation, if I only 00:29:18.350 --> 00:29:23.170 have one A in my sample, this probability of this happening 00:29:23.170 --> 00:29:24.320 is 0, right? 00:29:24.320 --> 00:29:27.390 Because I need two A's for this to happen. 00:29:27.390 --> 00:29:35.030 So really, the equation for this case is n A times n A minus 1. 00:29:35.030 --> 00:29:41.720 So it's k A plus A goes to B times n A times n A minus 1. 00:29:44.724 --> 00:29:47.140 And this is crucial for, like, the emulsion polymerization 00:29:47.140 --> 00:29:50.440 case, 'cause the fact that I don't have the second one is 00:29:50.440 --> 00:29:51.444 really important. 00:29:54.312 --> 00:29:56.270 Otherwise I always miscompute that this radical 00:29:56.270 --> 00:29:57.811 could react with itself and terminate 00:29:57.811 --> 00:29:58.906 itself, which is not true. 00:30:01.774 --> 00:30:03.100 Is that all right? 00:30:09.600 --> 00:30:13.260 I commented that these k's are a little bit odd. 00:30:13.260 --> 00:30:15.840 So if you remember, if you have a k for a unimolecular 00:30:15.840 --> 00:30:18.692 reaction, like C goes to A plus B, what units would it have? 00:30:22.550 --> 00:30:26.157 What are the units for regular kinetics? 00:30:26.157 --> 00:30:27.240 AUDIENCE: Inverse seconds. 00:30:27.240 --> 00:30:28.240 WILLIAM GREEN: Right, inverse seconds, right? 00:30:28.240 --> 00:30:29.740 So you would normally say this thing 00:30:29.740 --> 00:30:31.347 has units of inverse seconds. 00:30:31.347 --> 00:30:32.680 And that works fine here, right? 00:30:32.680 --> 00:30:36.760 Because I have dP dt, and it's in for a seconds times P, 00:30:36.760 --> 00:30:38.490 so it's fine. 00:30:38.490 --> 00:30:42.020 However, this guy-- to make the units work out, still 00:30:42.020 --> 00:30:44.452 has to also be inverse seconds, but that's not normally 00:30:44.452 --> 00:30:46.410 what you would have for a bimolecular reaction. 00:30:49.640 --> 00:30:54.020 And the other thing to watch out for here is that I have n's. 00:30:54.020 --> 00:30:56.270 But normally you would write them with concentrations, 00:30:56.270 --> 00:30:57.080 right? 00:30:57.080 --> 00:30:58.850 You have the-- you'd would write something 00:30:58.850 --> 00:31:02.490 like k times the concentration of C. It's 00:31:02.490 --> 00:31:05.060 the normal way you write your equations, the right equations. 00:31:05.060 --> 00:31:06.450 But this is not the same, right? 00:31:06.450 --> 00:31:07.460 This is n. 00:31:07.460 --> 00:31:11.010 This is unitless. 00:31:11.010 --> 00:31:13.580 Now it turns out, in this case, actually, you 00:31:13.580 --> 00:31:16.030 can use exactly the same as k and it works. 00:31:16.030 --> 00:31:19.120 But for the bimolecular case, it definitely does not work. 00:31:19.120 --> 00:31:20.950 The units are wrong. 00:31:20.950 --> 00:31:23.890 And also this concentration thing 00:31:23.890 --> 00:31:27.020 implies that we think the rate really has-- 00:31:27.020 --> 00:31:32.770 suppose we have k A k B. I could rewrite this as k times n 00:31:32.770 --> 00:31:37.880 A over the volume, n B over the volume. 00:31:37.880 --> 00:31:40.279 Is that all right? 00:31:40.279 --> 00:31:42.320 Actually one thing I should warn you about-- that 00:31:42.320 --> 00:31:45.420 usually when I write this, I write moles per liter. 00:31:45.420 --> 00:31:48.020 So I have factors of 10 to the 23 between this and this, 00:31:48.020 --> 00:31:51.520 because these n's have got to be the individual molecules 00:31:51.520 --> 00:31:52.780 for this to work. 00:31:52.780 --> 00:31:54.210 So watch out. 00:31:54.210 --> 00:31:57.680 First of all, there's a 10 to the 23 somewhere in here, 00:31:57.680 --> 00:31:58.180 right-- 00:31:58.180 --> 00:31:59.550 Avogadro's number. 00:31:59.550 --> 00:32:01.739 And then this is what-- 00:32:01.739 --> 00:32:03.030 how we would normally write it. 00:32:03.030 --> 00:32:06.360 But this k has the wrong units for us here. 00:32:06.360 --> 00:32:17.218 And so this guy here is the normal k divided by the volume. 00:32:22.008 --> 00:32:25.422 I think that's right. 00:32:25.422 --> 00:32:28.270 Maybe it's the other way around. 00:32:28.270 --> 00:32:31.489 No, this is normal k times the volume. 00:32:31.489 --> 00:32:32.030 That's right. 00:32:32.030 --> 00:32:37.450 So normally we would write units k normal-- 00:32:37.450 --> 00:32:47.430 normal kinetics, macro-- k for A plus B reactions, 00:32:47.430 --> 00:32:52.170 the units are centimeters cubed per mole second. 00:32:55.750 --> 00:32:57.385 But we want them to be units of-- 00:33:02.180 --> 00:33:08.227 the new k has got to be units of per second. 00:33:11.150 --> 00:33:23.515 And so the new k is equal to the old k new old times the volume. 00:33:26.160 --> 00:33:28.230 Is that right-- no divided by the volume. 00:33:28.230 --> 00:33:29.760 I was right the first time. 00:33:29.760 --> 00:33:32.810 This is volume-- yeah. 00:33:32.810 --> 00:33:36.530 Thank you-- centimetres cubed, centimeters cubed-- 00:33:36.530 --> 00:33:38.650 yeah, right. 00:33:38.650 --> 00:33:41.230 Cancelled out, good for a second. 00:33:41.230 --> 00:33:46.458 So this is k per volume normal k divided by n-- correct. 00:33:49.469 --> 00:33:51.260 And you may wonder, well how come it's only 00:33:51.260 --> 00:33:53.240 volume to the first power? 00:33:53.240 --> 00:33:54.880 Before I had volume to the second power 00:33:54.880 --> 00:33:56.930 in the denominator. 00:33:56.930 --> 00:34:01.480 And it's because I'm using n's up in the numerator. 00:34:01.480 --> 00:34:02.590 And it's just like here. 00:34:02.590 --> 00:34:05.216 I'm using n's instead of n over v. 00:34:05.216 --> 00:34:07.340 So I've sort of multiplied through by v everywhere. 00:34:07.340 --> 00:34:10.190 That's one way to look at it. 00:34:10.190 --> 00:34:16.248 All right, no questions yet? 00:34:16.248 --> 00:34:17.240 Yeah? 00:34:17.240 --> 00:34:18.728 AUDIENCE: So when you [INAUDIBLE] 00:34:21.219 --> 00:34:23.739 WILLIAM GREEN: Yeah, so I need a 10 to the 23 as well. 00:34:23.739 --> 00:34:26.180 So there's got to be an Avogadro's number. 00:34:26.180 --> 00:34:28.679 And you'll have to help me where the Avogadro's number goes. 00:34:28.679 --> 00:34:30.449 But it's got to be in there somewhere, 00:34:30.449 --> 00:34:34.597 to get from moles to molecules. 00:34:34.597 --> 00:34:36.055 So the numbers are a lot different. 00:34:41.005 --> 00:34:44.280 Is this OK? 00:34:44.280 --> 00:34:46.499 And this factor, this n minus 1 thing-- 00:34:46.499 --> 00:34:48.540 again, when these are macroscopic numbers like 10 00:34:48.540 --> 00:34:50.331 to the 23, this makes no difference, right? 00:34:50.331 --> 00:34:51.439 Because it's minus 1. 00:34:51.439 --> 00:34:53.775 So we can fit our data ignoring the 1-- 00:34:53.775 --> 00:34:55.400 It's 1. 00:34:55.400 --> 00:34:58.076 Perfectly well, the law of mass action-- it works fine. 00:34:58.076 --> 00:34:59.700 So you never notice this unless you get 00:34:59.700 --> 00:35:01.080 down to where n is very small. 00:35:04.940 --> 00:35:06.840 But this is actually the truth. 00:35:06.840 --> 00:35:08.999 This is really what the correct equation is. 00:35:14.870 --> 00:35:18.562 OK so now we have these-- the true equations 00:35:18.562 --> 00:35:20.520 for predicting how the probability distribution 00:35:20.520 --> 00:35:22.680 evolves in a reacting system. 00:35:22.680 --> 00:35:24.720 By the way, you can add transport in 00:35:24.720 --> 00:35:26.040 as well if you want. 00:35:26.040 --> 00:35:27.960 So you have control volumes, and you 00:35:27.960 --> 00:35:30.660 know formulas for the rate at which stuff is transported in 00:35:30.660 --> 00:35:32.220 and out of the control volume. 00:35:32.220 --> 00:35:33.870 You can add those terms in as well. 00:35:33.870 --> 00:35:37.240 And they'll affect the probability distribution. 00:35:37.240 --> 00:35:40.060 No problem, you should be able to do the same conversion. 00:35:40.060 --> 00:35:42.381 So just-- we've converted the reaction equations. 00:35:42.381 --> 00:35:43.630 You can write a similar thing. 00:35:43.630 --> 00:35:46.110 Once you get right the macroscopic transport equation, 00:35:46.110 --> 00:35:47.860 you can figure out how you would reduce it 00:35:47.860 --> 00:35:51.550 down to a time constant for transport 00:35:51.550 --> 00:35:53.660 that would work for the-- 00:35:53.660 --> 00:35:55.704 for this microscopic version of the equations. 00:35:55.704 --> 00:35:58.120 So transport reaction-- you can write the whole thing down 00:35:58.120 --> 00:35:58.994 as a master equation. 00:35:58.994 --> 00:36:01.390 That's a real solution, the real equation. 00:36:01.390 --> 00:36:04.300 And the problem is that the real equation oftentimes 00:36:04.300 --> 00:36:08.650 is too big, because P scales up so rapidly with the number 00:36:08.650 --> 00:36:10.562 of possible states. 00:36:10.562 --> 00:36:13.020 So then we have to figure out how are we going to solve it. 00:36:13.020 --> 00:36:14.740 And the way to solve it is by-- 00:36:14.740 --> 00:36:16.240 called kinetic Monte Carlo. 00:36:28.020 --> 00:36:33.090 And this method was invented by a guy named Joe Gillespie. 00:36:33.090 --> 00:36:34.755 And the journal paper is posted. 00:36:42.400 --> 00:36:45.140 And this is just like with Monte Carlo-- 00:36:45.140 --> 00:36:47.420 Metropolis Monte Carlo that you just did. 00:36:47.420 --> 00:36:49.930 We're trying to find a way to sample 00:36:49.930 --> 00:36:53.260 from the true probability density 00:36:53.260 --> 00:36:57.940 without ever computing the true probability density. 00:36:57.940 --> 00:36:59.760 So in the Monte Carlo integration 00:36:59.760 --> 00:37:01.450 you're doing for homework, you're 00:37:01.450 --> 00:37:04.905 sampling from the true Boltzmann probability density, 00:37:04.905 --> 00:37:06.530 but you never actually supposedly write 00:37:06.530 --> 00:37:09.665 down what that P is, right? 00:37:09.665 --> 00:37:11.540 And so we're going to do the same thing here, 00:37:11.540 --> 00:37:12.620 is we're going to-- 00:37:12.620 --> 00:37:15.147 from the differential equations-- 00:37:15.147 --> 00:37:16.730 from this differential equation system 00:37:16.730 --> 00:37:20.570 here, we're going to try to somehow never ever solve this 00:37:20.570 --> 00:37:24.144 directly, but instead just sample from the solution, P, 00:37:24.144 --> 00:37:25.060 which depends on time. 00:37:28.030 --> 00:37:34.070 And the way it's done is really to try 00:37:34.070 --> 00:37:36.980 to follow individual trajectories of what 00:37:36.980 --> 00:37:40.130 could have happened, OK? 00:37:40.130 --> 00:37:43.850 So at any time-step, we have some number 00:37:43.850 --> 00:37:46.510 of A's, B's and C's. 00:37:46.510 --> 00:37:49.600 And we're going to say, well in the next time 00:37:49.600 --> 00:37:52.700 period, some small time period, what could happen? 00:37:52.700 --> 00:37:54.700 And we're going to make the time period so small 00:37:54.700 --> 00:37:57.290 that the only thing that can happen is nothing happens, 00:37:57.290 --> 00:38:00.160 and A, B and C stay the same, or one event 00:38:00.160 --> 00:38:05.000 happens which will change one of the numbers, the A's, B's, 00:38:05.000 --> 00:38:08.870 and C's, by the same one reaction amount, OK? 00:38:08.870 --> 00:38:14.330 And then after we accomplish that, now we have a new state-- 00:38:14.330 --> 00:38:15.784 A, B, and C with some new values. 00:38:15.784 --> 00:38:17.200 And we'll do the same thing again. 00:38:17.200 --> 00:38:18.783 And we just repeat that over and over. 00:38:18.783 --> 00:38:19.990 So it's very brute force. 00:38:19.990 --> 00:38:25.250 You're just doing, like, what a single system would do. 00:38:25.250 --> 00:38:27.650 So for example, for the uranium atom case, 00:38:27.650 --> 00:38:29.690 we have one uranium atom in there. 00:38:29.690 --> 00:38:32.120 We wait for some time period, say 20 minutes. 00:38:32.120 --> 00:38:34.370 We compute the probability that the uranium would have 00:38:34.370 --> 00:38:36.130 decayed in the last 20 minutes. 00:38:36.130 --> 00:38:38.400 It's pretty small. 00:38:38.400 --> 00:38:42.490 And then we'll say, OK, now we'll draw a random number. 00:38:42.490 --> 00:38:47.840 If the random number is less than that probability, 00:38:47.840 --> 00:38:51.195 then we think that the uranium is still there. 00:38:51.195 --> 00:38:53.570 And if the random number is bigger than that probability, 00:38:53.570 --> 00:38:56.710 then the uranium is gone. 00:38:56.710 --> 00:38:59.370 If it's still there, then we'll do it again. 00:38:59.370 --> 00:39:01.690 And we'll just keep doing that over and over again. 00:39:01.690 --> 00:39:03.470 And we'll sample what could happen. 00:39:03.470 --> 00:39:07.650 The uranium atom could live for 38 years and 14 minutes, 00:39:07.650 --> 00:39:09.141 and then disappear. 00:39:09.141 --> 00:39:10.390 And that would be one example. 00:39:10.390 --> 00:39:11.973 And then we go back and we do it again 00:39:11.973 --> 00:39:13.590 from a different trajectory. 00:39:13.590 --> 00:39:15.420 And we do it over and over and over again. 00:39:15.420 --> 00:39:17.250 And we do it over and over again. 00:39:17.250 --> 00:39:20.130 Those trajectories are sampling from what could happen, 00:39:20.130 --> 00:39:21.935 with the right probability density. 00:39:21.935 --> 00:39:23.924 Does this make sense? 00:39:23.924 --> 00:39:25.340 So the way Gillespie says to do it 00:39:25.340 --> 00:39:44.012 is figure out the expected time until something would happen. 00:39:47.679 --> 00:39:48.720 We'll call this time tau. 00:39:53.090 --> 00:40:02.404 And we'll pull-- actually let's not use tau, 00:40:02.404 --> 00:40:03.820 because that's the only one that-- 00:40:03.820 --> 00:40:05.830 I'll try to be consistent with Joe's notation. 00:40:05.830 --> 00:40:07.510 He calls this thing 1/a. 00:40:20.925 --> 00:40:25.650 So a is like the expected rate per second. 00:40:25.650 --> 00:40:28.870 And 1/a is the expected time until something would happen. 00:40:28.870 --> 00:40:42.450 And then we'll draw a random number r-- 00:40:42.450 --> 00:40:43.220 he calls it r1. 00:40:46.867 --> 00:40:48.580 So we just call the rand function. 00:40:48.580 --> 00:40:51.480 It gives me a number between 0 and 1. 00:40:51.480 --> 00:40:55.430 And I'll say that my time-step, delta t-- 00:40:55.430 --> 00:41:02.100 which Joe calls tau, is 1/a the logarithm of 1/o. 00:41:06.790 --> 00:41:13.838 So I think there should be a negative sign. 00:41:13.838 --> 00:41:15.290 No, 1/r is positive. 00:41:15.290 --> 00:41:17.030 That's right-- positive. 00:41:17.030 --> 00:41:21.490 All right, so I choose a random number. 00:41:21.490 --> 00:41:22.790 Suppose it's 1/2. 00:41:22.790 --> 00:41:25.475 Then this fraction is 2-- 00:41:25.475 --> 00:41:27.080 1 over 1/2 is 2. 00:41:27.080 --> 00:41:29.780 Logarithm of 2 is some number bigger than 1. 00:41:29.780 --> 00:41:31.070 And I pull that. 00:41:31.070 --> 00:41:31.835 And I get some-- 00:41:34.540 --> 00:41:36.940 actually, some number less than 1-- logarithm of 2-- 00:41:36.940 --> 00:41:38.530 anyway, some positive number. 00:41:38.530 --> 00:41:42.040 Multiply it by the expected time until something happens-- 00:41:42.040 --> 00:41:44.410 and that's the time, in this particular simulation, 00:41:44.410 --> 00:41:46.725 when the next thing happened. 00:41:46.725 --> 00:41:48.850 And then I repeat this again, and again, and again. 00:41:48.850 --> 00:41:51.640 And I get the period of time intervals between when 00:41:51.640 --> 00:41:52.607 something happened. 00:41:52.607 --> 00:41:54.940 Now every time something happened, I have to figure out, 00:41:54.940 --> 00:41:56.930 well, what happened? 00:41:56.930 --> 00:42:00.626 So if I look at my-- 00:42:03.888 --> 00:42:07.360 plot back here when I had the uranium, I waited a long-- 00:42:10.090 --> 00:42:13.140 in this particular case, here's my tau. 00:42:13.140 --> 00:42:15.595 I'll call this tau 1-- 00:42:15.595 --> 00:42:18.232 the time that it took for the first thing to happen. 00:42:18.232 --> 00:42:19.690 And then, well-- what could happen? 00:42:19.690 --> 00:42:23.580 The only thing can happen with my single uranium atom case 00:42:23.580 --> 00:42:25.520 is it decayed. 00:42:25.520 --> 00:42:27.430 So it went away. 00:42:27.430 --> 00:42:30.250 All right, and then I do the random thing again. 00:42:30.250 --> 00:42:30.790 And I draw. 00:42:30.790 --> 00:42:32.415 And I get a different number this time. 00:42:34.234 --> 00:42:35.900 How long it is 'till something happened. 00:42:39.430 --> 00:42:42.090 Now notice-- maybe it's not so obvious-- 00:42:42.090 --> 00:42:46.535 that in the time until something happens changes as you run. 00:42:46.535 --> 00:42:50.840 So for example, if I started out with 100 uranium atoms 00:42:50.840 --> 00:42:52.070 in 100 little boxes. 00:42:52.070 --> 00:42:53.480 And I'm watching them. 00:42:53.480 --> 00:42:55.430 After I've run this for a long time, 00:42:55.430 --> 00:42:57.651 I only have 50 uranium atoms left. 00:42:57.651 --> 00:42:59.150 And that means that the average time 00:42:59.150 --> 00:43:01.482 period until something happens will be slower, 00:43:01.482 --> 00:43:02.690 because I don't have as many. 00:43:02.690 --> 00:43:06.060 The more I have, the more rapidly something will happen. 00:43:06.060 --> 00:43:07.660 Does that make sense? 00:43:07.660 --> 00:43:11.450 So I have to compute this a thing at each iteration. 00:43:11.450 --> 00:43:14.976 So I compute the time until something happens. 00:43:14.976 --> 00:43:16.950 Then I'm going to compute what happened. 00:43:16.950 --> 00:43:18.060 I'll tell you what that the second. 00:43:18.060 --> 00:43:20.185 In the uranium case, the only thing that can happen 00:43:20.185 --> 00:43:22.310 is one of the uranium atoms decayed. 00:43:22.310 --> 00:43:23.780 So I took that away. 00:43:23.780 --> 00:43:26.520 Now I have to compute a again. 00:43:26.520 --> 00:43:32.170 So a is the sum of all the things that can happen. 00:43:32.170 --> 00:43:34.980 So in my case where I have 100 uranium atoms, 00:43:34.980 --> 00:43:36.420 I have 100 different boxes. 00:43:36.420 --> 00:43:39.690 In each one of them, there is a rate at which uranium atom 00:43:39.690 --> 00:43:41.870 probability is decaying. 00:43:41.870 --> 00:43:44.480 It's like 1 over the normal half life uranium-- or something 00:43:44.480 --> 00:43:45.690 like that. 00:43:45.690 --> 00:43:48.310 And I add them up. 00:43:48.310 --> 00:44:01.190 So a is the sum of the rates of everything that can happen. 00:44:10.017 --> 00:44:11.850 So we just list out all the different things 00:44:11.850 --> 00:44:13.240 that can happen. 00:44:13.240 --> 00:44:15.330 So in the case-- suppose I have two boxes-- 00:44:15.330 --> 00:44:17.160 two uranium atoms. 00:44:17.160 --> 00:44:20.220 I can have decay of uranium atom number one or uranium atom 00:44:20.220 --> 00:44:21.630 number two. 00:44:21.630 --> 00:44:25.394 So I have-- this thing would be equal to-- 00:44:25.394 --> 00:44:26.600 so two uranium atoms-- 00:44:33.500 --> 00:44:40.914 a is equal to 1 over tau normal for a uranium atom plus one 00:44:40.914 --> 00:44:41.580 over tau normal. 00:44:41.580 --> 00:44:44.700 Because they're both normal uranium atoms. 00:44:44.700 --> 00:44:47.040 So it's equal to 2/tau. 00:44:47.040 --> 00:44:50.620 And so 1/a is half of the normal lifetime. 00:44:50.620 --> 00:44:53.430 And that's when I would expect that something 00:44:53.430 --> 00:44:57.570 would happen by then, OK? 00:44:57.570 --> 00:45:00.550 One of them would probably decay. 00:45:00.550 --> 00:45:05.050 So that would be the a value I would use here. 00:45:05.050 --> 00:45:07.230 Draw my random number, get my tau 00:45:07.230 --> 00:45:09.480 until something probably happened, 00:45:09.480 --> 00:45:11.400 then figure out what happens. 00:45:11.400 --> 00:45:13.830 Now in the kinetics case, several different things 00:45:13.830 --> 00:45:14.580 can happen, right? 00:45:14.580 --> 00:45:23.275 I wrote down if I'm in state n A n B n C, I-- 00:45:23.275 --> 00:45:26.850 the state can change by two different ways. 00:45:26.850 --> 00:45:33.330 I can move out of that state this way or this way. 00:45:33.330 --> 00:45:34.830 Two different reactions can happen-- 00:45:34.830 --> 00:45:36.470 a forward or reverse reaction. 00:45:36.470 --> 00:45:38.370 And they would both take me out of the state. 00:45:38.370 --> 00:45:40.453 So those are two different things that can happen. 00:45:42.930 --> 00:45:46.680 And so I have to add those rates. 00:45:46.680 --> 00:45:54.510 So I would add this number and this number. 00:45:54.510 --> 00:45:56.970 And they would be the two things that could happen. 00:45:56.970 --> 00:45:59.740 And that would give me the a value. 00:45:59.740 --> 00:46:03.690 Now I figured out something happened. 00:46:03.690 --> 00:46:04.940 But now I'm not doing uranium. 00:46:04.940 --> 00:46:07.649 Now doing, say, number of A's. 00:46:07.649 --> 00:46:09.190 Now I figured out something happened. 00:46:09.190 --> 00:46:10.981 But now I have to figure out what happened. 00:46:10.981 --> 00:46:14.080 And the number of A's could have gone up or gone down, 00:46:14.080 --> 00:46:15.730 because if it was a forward reaction, 00:46:15.730 --> 00:46:16.997 the number of A's went down. 00:46:16.997 --> 00:46:19.330 If it was a reverse reaction, the number of A's went up. 00:46:19.330 --> 00:46:22.540 So I have to decide which way is going to happen. 00:46:22.540 --> 00:46:25.780 So I could have either the number of A's went down, 00:46:25.780 --> 00:46:27.850 or the number of A's went up. 00:46:27.850 --> 00:46:29.150 Those are two possibilities. 00:46:29.150 --> 00:46:30.820 I have to figure out which one happened. 00:46:30.820 --> 00:46:31.590 I have to figure-- 00:46:31.590 --> 00:46:33.670 draw another random number to figure out 00:46:33.670 --> 00:46:35.870 which one probably happened. 00:46:35.870 --> 00:46:38.590 So I'll list out the list of all the things that 00:46:38.590 --> 00:46:42.210 can happen with their rates. 00:46:42.210 --> 00:46:46.350 And one of them, say, is twice as big as the other. 00:46:46.350 --> 00:46:49.000 So that'll be twice as likely to happen. 00:46:49.000 --> 00:46:50.880 So I have to choose a random number, 00:46:50.880 --> 00:46:53.220 and use that to choose the next one. 00:46:53.220 --> 00:47:00.722 So first I figured out-- 00:47:00.722 --> 00:47:02.180 this is actually what we did first. 00:47:02.180 --> 00:47:04.700 I figured out what my rates are. 00:47:04.700 --> 00:47:07.922 Second thing is I figure out the time until something happened. 00:47:07.922 --> 00:47:10.130 Third thing is I'm going to figure out what happened. 00:47:16.030 --> 00:47:18.240 And what I do is I say-- 00:47:18.240 --> 00:47:21.290 I draw a random number called r2. 00:47:25.079 --> 00:47:26.620 And if there's only two possibilities 00:47:26.620 --> 00:47:29.280 like in this case, then I can say 00:47:29.280 --> 00:47:43.690 if r2 is less than the rate of process one divided 00:47:43.690 --> 00:47:49.200 by the sum of the rates, which is the same as the a, right? 00:47:49.200 --> 00:47:53.810 So this is-- it this is true, then process one happened. 00:47:57.769 --> 00:48:00.310 On the other hand, if the rate-- if the random number I chose 00:48:00.310 --> 00:48:05.206 is larger than this ratio, then I'll say process two happened. 00:48:05.206 --> 00:48:07.830 And that's how I decide whether a number of A's went up or went 00:48:07.830 --> 00:48:09.717 down. 00:48:09.717 --> 00:48:12.300 And then I would recompute the a-- depending on what happened. 00:48:12.300 --> 00:48:13.490 Suppose I do this. 00:48:13.490 --> 00:48:16.294 And I found out that the number of A's went up-- 00:48:16.294 --> 00:48:19.100 because I-- by drawing the random number. 00:48:19.100 --> 00:48:21.905 Then I'll go and I'll recompute the A. Now 00:48:21.905 --> 00:48:24.430 the rates will all be different, because n A is bigger. 00:48:24.430 --> 00:48:27.050 So the rate of the n A reaction is faster. 00:48:27.050 --> 00:48:28.320 So I recompute it. 00:48:28.320 --> 00:48:33.170 I redraw a number here, and get some different time, tau 2. 00:48:33.170 --> 00:48:35.290 It's not the same as tau 1. 00:48:35.290 --> 00:48:36.790 And then something happened here. 00:48:36.790 --> 00:48:39.012 I do the same process here out what happened. 00:48:39.012 --> 00:48:41.220 Maybe this time A reacted away and it went back down. 00:48:44.310 --> 00:48:46.810 And then I do it again. 00:48:46.810 --> 00:48:50.190 And this time I draw a longer time, tau 3. 00:48:50.190 --> 00:48:51.510 And I do-- what happened? 00:48:51.510 --> 00:48:53.635 And maybe this time, again, something reacted away. 00:48:53.635 --> 00:48:55.100 And it went down. 00:48:55.100 --> 00:48:57.320 And on, and on, and on-- 00:48:57.320 --> 00:48:59.710 and after my computer is done doing that for a while, 00:48:59.710 --> 00:49:02.739 like, after a long enough time, I say, OK, I'm done. 00:49:02.739 --> 00:49:03.780 Oh-- I'm not really done. 00:49:03.780 --> 00:49:05.321 I have to go back and start it again, 00:49:05.321 --> 00:49:08.540 and run another simulation to run 00:49:08.540 --> 00:49:10.210 through all the possibilities. 00:49:10.210 --> 00:49:12.230 And I want to save all these results of all 00:49:12.230 --> 00:49:14.150 these simulations, because these guys are 00:49:14.150 --> 00:49:17.077 samples from the probability distribution. 00:49:17.077 --> 00:49:19.160 And if I built histograms of stuff from those guys 00:49:19.160 --> 00:49:21.336 I can figure out what really is going to happen. 00:49:21.336 --> 00:49:26.188 All right, so it's a way to represent the P. Questions? 00:49:29.954 --> 00:49:31.120 All right-- perfectly clear? 00:49:33.770 --> 00:49:38.540 All right, so what's nice about this is actually, 00:49:38.540 --> 00:49:41.032 it's not that hard once you do this once. 00:49:41.032 --> 00:49:41.990 It's not hard to do it. 00:49:41.990 --> 00:49:44.280 And you can do it for any problem at all. 00:49:44.280 --> 00:49:45.700 And so it's like-- 00:49:45.700 --> 00:49:47.990 often cases people do this even for cases 00:49:47.990 --> 00:49:51.260 where you actually could solve-- 00:49:51.260 --> 00:49:53.267 you might be able to solve this equation. 00:49:53.267 --> 00:49:54.600 A lot times, people won't do it. 00:49:54.600 --> 00:49:57.882 They'll do this instead just because it's so easy. 00:49:57.882 --> 00:49:59.840 So though maybe it looks hard to you right now, 00:49:59.840 --> 00:50:00.990 it's only a three step thing. 00:50:00.990 --> 00:50:02.330 And just do it over, and over again. 00:50:02.330 --> 00:50:02.870 You're not doing it. 00:50:02.870 --> 00:50:04.130 The computer's doing it-- 00:50:04.130 --> 00:50:05.580 no big deal. 00:50:05.580 --> 00:50:07.164 And so this one you have to, like, 00:50:07.164 --> 00:50:08.830 look at the sparsity pattern of your m-- 00:50:08.830 --> 00:50:11.390 and some complicated stuff. 00:50:11.390 --> 00:50:16.210 So this is actually done extremely commonly. 00:50:16.210 --> 00:50:19.260 All right, see you on Friday.