 So one thing I should say, there was one problem that you should just turn it next time with the last hallmark in chapter 7 about diffusion. So let's see, does anybody know about this? Looks like there's a spot 80 miles wide of oil happening and I think they tried to burn it or something. Yeah, the oil rig that was damaged. And I think this is a kind of a prediction of what it will happen. Well, today's already 28, so. Okay, we're not going to talk about this complicated process. A much simpler one, but... And then guess what? There is actually another... This thing seems to be in the news a lot, right? This is the Icelandic volcano that I'm trying to see what the... I was trying to find some kind of an in-time, like a movie or animation of how it evolved, but it evolved very small and then kind of it spread all over, right? I think it even reached the US airspace. And again, this is quite irregular, right? And of course it's quite irregular because the wind patterns are all irregular and all that. But I think the basic concept that's underlying this type of processes is diffusion. Diffusion of some particles in some medium, right? So, we'll briefly talk about this because it's connected to essentially my theorem. And then we'll move to chapter eight. So, this is by the way kind of totally disjoint subject. So, all right, so let me start by talking a little bit about the diffusion, which simply refers to diffusion of small particles of some say pollutant or chemical in a medium. And the assumption is that these small particles kind of move randomly. So, individually they perform what's known as Brownian motion. Now, Brownian motion was invented a long time ago, right? So, Brown was, I think, a biologist in the 1900s, right? That studied, I think, the pollen particles. And he observed that there is this random fluctuations. And there is, you know, individual particles kind of follow a random walk or a random pattern, right? Random positions. The jumps are kind of random. But on ensemble, if you look at the whole, you know, pollutant or chemical or pollen, then you start seeing the same kind of recurrent pattern that comes from the central limit theorem. That is, you have a normal distribution. If you start very concentrated at one location, then as those particles kind of displace or, you know, move randomly in the medium, after some time you see some sort of a Gaussian distribution. So, a normal distribution. Okay, then I think later, so this was originated with Robert Brown, actually early in the 19th century. Then Albert Einstein was actually, or Einstein, was very, very big, actually studied a lot of this diffusion of particles and fluids. So, again, I think he was trying to understand how fluid, you know, affects the motion of particles, like friction and other things. So, the fluid, you know, it wasn't like very small particle. It was rather large particles and it was starting to drag of those particles and fluids. But, of course, the mathematics started, or the mathematicians started paying attention in the 19th, in the early 20th century, 1923, with Norbert Wiener. Okay, so this is kind of an old subject and it's a very interesting subject in its own. It led basically to discrete models and also to continuous models. Discrete models are referred to random walks and, of course, continuous models would be the diffusion or PDE models, the partial differential equations. And random walks is an area of probability, okay? So, I don't want to say too much about random walks or discrete models. In this short section, sort of, we just talk about the PDE model, the diffusion equation, okay? So, I guess the only thing to say about the discrete model is in 1D, it's a very simple interpretation of this random walk. So, imagine you have a particle that starts at an initial time, say at zero, and it moves, so this is the particle. It moves only in discrete moment, you know, in discrete intervals of time, like after one unit of time, it can move only to designated sites. So, let's say the integers, zero, one, two, negative one, right, and so forth. So, particle position can be anywhere in integers, so zero plus or minus one, plus or minus two, okay? And it moves with, so it moves, so at each iteration, particle moves with equal probability to the left, so that is equal probability of one-half to the left side, neighboring side, or right neighboring side, okay? So, can imagine it moves here with probability one-half, or it can move here with probability one-half, and this is at every site. So, one realization of this experiment would follow this particle, right? As it moves, you know, let's say, you know, infinitely, an infinite amount of time. Up to time t, for instance, let's say, okay? Yeah, it doesn't stay still, it just always moves either left or right. Again, simplest kind of random walk. So, the question is, what happens if you start with lots of particles, all of them centered at zero? What is going to happen is that number of, as each of those particles is moving, you know, according to this random walk. So, if you have a high concentration, so if at time zero, you have a high concentration of particles, all at location zero, let's say, right? After time t, actually what you're going to see, you're going to see the standard normal distribution. Excuse me. There was a normal distribution, a Gaussian distribution. So, on a picture, it would be, if you have lots of particles, so we're talking about lots of particles and they're all independent of each other. So, let's say you have lots of particles stacked on top of each other here, right? Now, you let this person at time t greater than zero, what you're going to see is you're going to see a, well, in the limit, if you were to have infinitely many particles in the limit, you would actually see this distribution with mean zero and standard deviation, I believe, squared of t. So, the variance would be actually t. And this follows from the central limit theorem. So, it follows from the central limit theorem. Okay. By setting appropriately the random variables, right? So, x1, xn, right? Right, so x1 would be, so each x would be the, it would be that either plus one or minus one, right? Each would probably have. And then the sum x1 through xn would actually measure the location of that particle after time t. So, using the central limit theorem, you can actually bound that location. You can actually say the location is going to be within like two standard deviations with probably 95%, right? So, you can kind of fit into this, into this, okay? Now, as I said, I want to focus only on continuous models where instead of looking at individual particles, even if there are lots of them, we're looking at the concentration of the particles. So, if we call c of x and t to be the concentration of particles, which simply means mass divided by volume. So, the mass of the particles in a given volume of the medium, okay? And let's say we still talk about one dimensional for now. Then what you see is you see the following. So, on the x-axis, which would be the medium, right? At time t, you have, so we want to do some sort of a mass balance. We want to look at the amount of material, particles or materials. So, the mass in an interval, in an infinitesimal interval, x and x plus delta x. So, a small region of that medium at a time t and at a subsequent time t plus delta t, okay? So, if you think about particles possibly leaving this region, right? So, we're going to focus on this region. And now we're going to say, you know, it's possible that particles are moving away to left or to the right at some rate, okay? So, we're going to call denote q of x to be the rate at which particles or the mass, mass of the pollutant or chemical or whatever it is, the rate at which particles pass location x moving to the right. So, really it should be, I mean that's just a convention, right? So, it should be moving to the right. So, here I really want to put minus q, right? So, if it's moving to the left, we're going to say the rate at which is moving to the left is minus q, okay? And then, when you do the, okay, so now if you do this balance or conservation of number of particles, the conservation of mass basically says the concentration at the new time delta t equals the concentration at the old time. So, think about this as, well, maybe concentration times, I should say this, concentration times the length would be the mass, right? So, this is mass. This is a mass that's in that medium portion of the medium at time t plus delta t, okay? And we say that this mass is the mass that was at the previous time plus or actually minus, minus the mass that actually left the region through the right boundary, right? So, this is going to be minus, what's the mass? It's going to be q of x plus delta x and t times delta t. So, that's the mass leaving at right end, right? And then also, it's whatever is moving to the right from the left boundary which is q of x and t delta t. So, this is the mass entering at left end point. It's just kind of counting the mass that is at the next time step. So, this is the original mass. Okay? So, this is the new mass at time t plus delta t, okay? So, here's how we can write it, c of x t plus delta t minus c of x t. And I can divide by delta t and delta x. So, the only thing that's left here is delta t. And then on the other side is minus q of x plus delta x minus q of x t. And here's the only thing left is going to be delta x. Okay? As delta x and delta t go to zero, because remember these are supposed to be continuous models, meaning these changes have to be infinitesimally small, then you see that you get some partial derivatives. So, partial with respect to t of c equals minus partial with respect to x of q. So, this is what's known as a sort of conservation of mass in this simple one-dimensional model of particles moving randomly, right? In and out of this region or along a line, along in 1d. Okay? Now, on top of this law, conservation of mass, there is so-called fixed law, which says that there is a relation between the rate at which particles move across a boundary and the concentration gradient. So, that says that the rate at which particles move across x from left to right is proportional to the concentration gradient. So, and that's written as follows. So, q is going to be proportional, and there's going to be a constant of proportionality, which you'll see it's negative. And that's the gradient, concentration gradient is partial of c with respect to x at x and t. So, intuitively this simply says that if I have, if at a location I have a low gradient, okay, so the particles are concentrated here, less concentrated here, right? So, I have kind of a low derivative that is small, right? Negative but small. This means that the particles will actually move to the right, right? They'll try to kind of get uniform, so this is low concentration gradient. Whereas, if it's a high concentration gradient at a location, then what happens? The rate at which particles will actually move is going to be, well, accordingly higher, right? So, in that sense it's, we say that it's proportional to each other. I do have, well, there are a bunch of applets, I think if I can show you just one of them. So, if you have, of course this is actually now in 2D, but if you see a high concentration, let's say of these particles, then there is sort of a high, you know, the rate at which these particles cross that point is actually going to be higher than what it is now, right? So, this, now the concentration is kind of low, and again imagine these are like not just 20 or 30 dots, but lots of them, right? Then the concentration is going to be the rate at which the flux, what do we say the flux, q is going to be proportional to the concentration gradient. Okay, so these two things, one on top of each other, give you, you see you can actually plug in the q into the first equation and you get what's called a PDE. D is a constant here, well, it's not always a constant, but D is called a diffusivity constant. And it may depend on the medium, on the pollutants, you know, it can depend on various factors. And it could be not, it also is possible that it's not constant, but in this, let's just assume to be constant. So, what you get is you get, partially with respect to t of c, yep, is minus partially with respect to x of q, which is minus, minus d over 2, I should put minus partially with respect to x, minus d over 2, partially with respect to x of c. So what you get is you get the following, assuming that D is constant, the standard diffusion equation, that the concentration actually satisfies. Concentration satisfies. So this is kind of a, it's one of the most important PDE examples, partial differential equations, right? Now, how to solve partial differential equations actually takes a whole course. Even this simple differential equation, PDE, can be solved in many different, you know, several different ways. So, instead of going through that, I think the book talks a little bit about using Fourier transform, okay? So if you're familiar with Fourier transform, you know, it's okay to, I mean, probably you've seen this already. But if you haven't, I don't want to spend too much time. I'll just say, you know, I'll just give you sort of what you've seen in the, even in the ordinary differential equation course, when somebody gives you an equation and gives you a solution and says, check that that solution satisfies that equation. Okay. It's not really solving the equation, but, again, due to the limitation of time, that's probably the easiest way to do it. So it turns out that one can check, whoops, that the following is actually a solution. So it's 1 over square root of 2 pi d times t e to the minus x square over 2 dt is a solution of the diffusion equation. Okay. And how do you check this? You just differentiate, right? Now, you see, when you differentiate with respect to t, it's already a product here, so you have to do a little bit of work. It's not just say, look at it, and you see it. Also, when it's with respect to x, you have to differentiate it twice. So the first time you differentiate, you're going to get x times some exponential. So the second time you differentiate it again is going to be a product. Okay. But you must do this. If you've never done this, you must do it. Okay. Just to verify that this is a solution. So simply just show that derivative with respect to t is equal to the second derivative with respect to x with that constant in front d over 2. Okay. Now, there is the fundamental fact about this solution is that if you look for other solutions, you won't find other solutions of this equation. Again, assuming you're looking on the whole line. So this is actually pretty much the only solution of this equation except a constant multiple of this. So any solution, I shouldn't say that any solution, but all the solutions can be kind of referred back to. So more general, we will talk about this constant multiple of this solution as being the fundamental solution of this equation. Okay. And let me tell you why. So if you are to, well, obviously t cannot be zero. So t has to be positive so you can divide by t and also take the square root. But if t is very close to zero, so very close to zero, this graph is simply the bell shaped curve in x with respect to x. And at x equals zero, what is the height? e to the zero is one, right? So the height, the concentration in the middle is p naught over the square root, right? So if t is very small, that's going to be very large, right? So it's going to be something like this. So it's going to be very concentrated at x equals zero. And again, this goes to infinity as one over square root of t. As t gets larger and larger, so maybe the next t is, let's say, of the order of one, then it's going to look like a Gaussian. Yeah. And again, it's going to have a peak that, of course, is going to depend on this p naught. Okay. So the peak is going to be p naught over square root of 2 pi square root of dt. And if t is large, what's going to happen? It's going to be very small. The peak is going to be very small, but it's going to be very wide here, right? So in fact, what's the best way to talk about this width, basically the standard deviation, right? Yeah, the standard deviation. So the width of this bulk of the pollutant or of chemical would be what? Well, if we consider this to be, well, this is Gaussian, we can think about the standard deviation as one standard deviation or two standard deviations from the mean, right? So what's going to be the standard deviation here? dt, right? Because remember it was x square over 2 sigma. Well, 2 sigma squared, right? So sigma squared dt. So you see that sigma is actually, as we said earlier, the standard deviation grows like squared of t. Yeah, the variance is dt. So the variance is linear in t and the standard deviation is squared. So this, the bulk of this material is going to expand, right? Or spread by not a linear rate, but as a square root of t, right? Of course, it's all assumed d is constant. It's a constant divisivity in the medium. So let's see. So the example is that mentioned in the book is of a, I think, chemical accident. So like airborne pollutant. So this is an example of pollutant following an accident at a chemical plant, right? So it says initially you have some sort of a high concentration of this pollutant that you kind of care about at location. This is where the plant is, right? And it's just, you know, it's like a concentrated, it's concentrated around this location. And then subsequently what happens is you have some, so I should say the concentration, if you were to plot it, would be kind of here, right? But subsequently you have, you notice this plume as being somewhere away from this, somewhat dispersed. And here's like a town or a city, okay? And you are kind of wondering what this pollutant is going to do as time evolves. Is it going to affect this town or not, okay? So what's known is that an hour after release, a toxic cloud that's 2,000 meters long, okay? So it's somewhere, okay, so after an hour this is 2 kilometers long plume. And the maximum concentration of pollutant in the cloud is 20 times the safe level. So the maximum concentration is 20 times the safe level. The question is what is the maximum concentration expected in town? And how long will it take until the concentration falls back below the safe levels, okay? So what happens here is that there is some drift, okay, some drift, some wind, which in absence of the wind you would have just the concentration be always centered at where the power plant is and just diffuse on both sides, right? But there is some, the wind speed is given by, so wind speed is given to be 3 kilometers per hour, which is very small actually, right? Okay, so how do you model this? Again, the very simple assumptions to make is that, let's see, we should have the distance from the plant to the, yeah, 10 kilometers upwind, okay? So we have this distance to be 10 kilometers from the plant to the town, right? So we'd like to know how this plume evolves and how will it be felt in town, right? Is it going to be above the safe level or below the safe level? One is going to be maximum and so forth. So the model is going to say the following. So it turns out that in the presence of wind, the PDE, and again we're not going to actually talk about derivation of this, but if there's no wind, this term doesn't show up, right? There's no term like that, but in the presence of wind, this term is added to that model, to that derivation. And the outcome of this is the following. It's saying that the solution is of the following form. And again, one can see this through the, through various, you know, just by direct computation, if you want, is the following. So it says that it's 1 over, or p naught, excuse me, over square of 2 pi dt, e to the minus, and this is x minus vt squared over 2 dt. So the only effect of this drift is that there's a shift in the peak, you know, in the location of the peak, okay? So you can actually check this if you want. You can check that when you take the derivatives, well, or you can make a change of variables to see that this term shows up. So with this term in, you have an equality of the derivatives. I think change of variable is the easiest. So the solution really looks, as we said, it looks, it starts very concentrated, right? Then it kind of drifts. So it starts at, if you want, at time zero, at location zero, but then it kind of starts diffusing and drifting, right? So this guy is going to be, this should be higher here, but the peak is actually moving, right? So this is the peak location. So the peak location moves with speed v, basically. Okay, so let's see. So how do you actually plug in the numbers here? So to say that at time t1, your concentration is peak concentration is 20 times the, what, the safe level, right? I'm honest to say the following. So you just say that c of zero and one, so at time one, this is time, time one. And, okay, so this zero is kind of relative. But it's just to say that the concentration is 20. And this equals, according to that formula, maybe I should say this is not at zero and one. But so at time one, you see already your peak is at location v. So maybe this should be v. Okay, this is 20. This is p0 divided by square root of 2 pi d times 1, okay? And this is e to the minus v minus v times 1. So this is zero, basically. e to the zero over 2 dt, or d times 1. So this is just one. At time one, the position is the velocity here. Or another way to say this, you can count the, you can take the reference, so if this is at time one, you can take the reference to be x0, it should be the one where the peak concentration is. But if you take x to be zero where the plant is, then it's just a shift here. I think maybe it's best to leave it where the plant is, x is zero. At time one, this is going to be x equals v, right? And in general, x is going to be v times t, okay? Anyway, so this is zero, so this just tells you that what this p0 is. So by the way, the reason for this, so you need this information to figure out what that constant is, because that constant is not, you know, is pretty much arbitrary unless you have this extra information. So 20 times, okay? So you see this constant still depends on the d, on that diffusivity. Diffusivity is going to come out of the width of the plume. Yeah, go ahead. Right, but okay, so the safe level is, whatever the safe level is, right? You can normalize, you can actually always refer to that as being, so it's always relative to that. So your safe level is assumed to be one, for instance. Yeah, you're right, it should be a constant, and then it would carry that constant all the way through. So the assumption is that it's always concentration relative to that level, yeah. Okay, so now what is d? Well, d comes from the width of the plume at time one, which is given to be two kilometers, right? Two kilometers, okay? It's two kilometers. And the assumption is that the width is actually four standard deviations. So why is that? Because within four standard deviations from the mean, so if you are two standard deviations to left or right of the mean, then you have 95% of the material, okay? It's within that. So that's the bulk of the plume, basically. Now, remember, in reality, the plume doesn't look just like a bell shape, okay? So this is just a crude approximation. If you're using this model, the point is that if you're using this model, this is a reasonable assumption to make. So it's four standard deviations. That is, the standard deviation is one fourth, right? So if the width is four sigma, then sigma has to be 0.5 kilometers. So what's the variance? Or what's going to be d, basically? At time one, it's going to be d times t, right? The variance is d times t, remember that? The variance is the square of the standard deviation is d times t. E in this model again. And this is just 0.25, okay? So this is the d, and that's what you can find the p0 to be from the previous one. P0 was 20 times, okay, 20 times. Now remember, never rush to plug in the numbers if you don't have to. But you can do it, right? So if you need p0. Okay, so why, oh, okay. So it turns out this is like close to 8 if you want to have kind of a rough idea of what p0 is. Okay? So now you can go back and say, well, now I know precisely what the concentration is at all times and at all locations. Right? It's going to be, what is it going to be? It's going to be this particular p0. So this is, can I write it as like 8? I shouldn't write it as 8. I should write it as this is p0 divided by the square root of 2 pi dt. e to the minus x minus vt squared over 2 dt. And the only thing that are floating in this formula are x and the t, right? Everything else is determined, okay? So the last thing is to say that the concentration in the town is when x is 10, and that's just a function of time and it's going to be what? 20 over square root of t e to the minus 10 minus 3t square over 0.5, v was 0.25t. Okay? So if you are just, you're an observer at that location and you see the, so your location is here, right? You see the wind. I mean, you see the thing coming at you. And you can see a small, very small concentration here. Then an increasing concentration. As this bloom kind of moves, it's going to move above you, but it's going to be at a much lower peak, right? Because as it moves, it kind of slows down, right? So that peak is given by this formula, that concentration at the location where you are. It follows this. You can plot it and see, right? So you can see the maximum and so forth. So let's see. So what is the, yeah, so you can maximize it. So now I can answer all the questions, like maximum concentration is going to be p max, which turns out to be, what is it? 10.97 at t max, which is about 3.3. And that's in hours, hours. So after 3.3 hours, you'll see the peak concentration. And it's going to be, this means 3.3 means it's above 1, so it means it's, no, I'm sorry, 10.97 means it's going to be 10 times the safe level, right? Now, what's the answer to that question? Like after how much time is going to go back to the safe levels, set this equal to 1 and solve for what? How many t's are you going to get? Two of them, right? Which one is going to pick? They're both going to be positive. The later value, right? Because the first value of t is going to be where you're first hit by this wave, right? You see when it's going to take a little bit of time and then you're going to, right? Initially there's your safe level, right? But then the first time is going to be the concentration equal to 1. It's going to be the start of the pollutant period. And then the later value is going to be when it's going to be back to safe, okay? Simple model, right? It's a very simple model, but it's still not obvious, okay? It's not, so the outcome of this is going to be something that you can go and tell somebody that cares about this, right? That has no idea about math, okay? So it falls within that kind of modeling process that we've been talking about. And the book also talks a little bit about sensitivity. Lots of assumptions here, right? One of the assumptions is certainly the speed of the wind. It's never constant, but even if it's constant, you may not know it. Like you can make an approximation. So you can see this. There's a table of values for different wind speeds, different high concentrations, the time until safe levels, right? And so forth, yeah. So the concentration of mass, what we love, is that the same for diffusion and fluids? Or does it change? Diffusion and fluids. Well, in different mediums, women? Yeah, in different mediums. The constant is going to change for sure, the diffusivity. Now, you also can have what's called anomalous diffusion. If you do have normal diffusion, then that's what normal diffusion is, okay? Now, if you do an experiment and you're able to measure something that contradicts this model, then you have to go back and say, I'm going to change my assumptions. One assumption is that things are following normal diffusion. So that's what's called anomalous diffusion. It's just kind of a revisit, or it's a change in the assumption. The assumption is that these are not just particles of pollen in the air, right? It's not springtime. It's actually a bad thing happening, so, right? It's an oil spill. I mean, for oil, certainly this is not normal diffusion. Why is it not normal diffusion? Well, oil particles don't mix with water, so they always try to stick together, right? So in fact, I don't even think it's appropriate to call it diffusion. It's more like dispersion, right? It disperses, right? It stays together. But it's just kind of spreads according to the wave patterns, wind patterns, current patterns. The volcano ash is more like diffusion, right? But it may not be normal diffusion. That is, as that plume spreads, let's say there's no wind. The standard deviation may not go like square root of t. So, anomalous diffusion is when this standard deviation is not like t to the one-half, but t to some power, or alpha is not one-half. Okay? And this actually has been observed in lots of physical things. For instance, so the author of this book has been at the University of Nevada, Reno. So they were doing water in porous media, so you have water in underground, right? That's diffusion, but it's certainly not normal diffusion, right? So there are some anomalous type of diffusions. And it's kind of an interesting thing. I mean, interesting thing is that to model such things, you end up with equations that look like this. And maybe I shouldn't scare you too much. But I don't know, this is where it's not the same alpha as this. Maybe I should call this better. It's where you have fractional order derivatives. So do you know what the half derivative of X is? Okay, so it's a whole new field. It's called fractional calculus. That's what it is. It's messy. So again, I think it's actually current area of research, fractional order, partial differential equations, or probabilistic anomalous diffusion processes. So there's that kind of thing. Let's assume normal, let's demonstrate that a little bit. But as always, you assume the simplest first. If you can get away with that and get a big box, you know, but if you're seeing some, I mean, if your model is going to be, you know, it may be valid for a day or two, and then, right? So it depends on the scope of the model. It's of the modeling process, right? So all of this have to be adjusted. There's always readjustments and reconsiderations of the assumptions. But again, you know, all we do here is just that the only sensitivity that one does is to sum up the parameters in this model, but the model is the same, right? So the parameters are... What's the wind speed? Wind speed. What else could it be? I think that's pretty much... The velocity of the beam if you're in water, so... D. Certainly D, yes. Yeah, so, okay. So actually, the author even talks about this anomalous diffusion. There is an extremely important work, especially in the west here because of the water shortage. This anomalous diffusion is huge. There's a group at Colorado School of Mines that does a lot of this research. Not in math, but in geosciences, you know. Okay, anyway, I thought it was kind of interesting that two things happened recently that have something to do with this. It's all about how that thing spreads, you know, and affects, you know, is it going to affect this airport, or is it going to affect this coastline, or is it going to affect... And you can see how much more complicated things are because of... This was boring, but this is not boring, right? Things are evolving in time, patterns change. Who knows? The next volcano is going to erupt. I need to go to Romania fairly soon, so keep your fingers crossed. But anyway, any questions? So we're going to have to leave Markov Change for next week. Chapter 8, yeah. What's supposed to be 16 for the next meeting? You certainly should do 16, and then the other two problems, three I think maybe will make it two only from Markov Change for the last homework, but certainly number 16 is very close to what we talked about, so... Okay? Say it again? It's posted already, but I'll just delete probably... We'll see how much we do Monday. Okay, so, but I would say do this for Monday. Okay, so I need to... Somebody to administer this, somebody turn it, okay. So this is how the FCQs...