title

The diffusion equation | Week 12 | MIT 18.S191 Fall 2020 | Grant Sanderson

description

How the diffusion equation can arise from a simple random walk model.

detail

{'title': 'The diffusion equation | Week 12 | MIT 18.S191 Fall 2020 | Grant Sanderson', 'heatmap': [{'end': 701.97, 'start': 539.097, 'weight': 0.774}, {'end': 723.035, 'start': 708.359, 'weight': 0.703}, {'end': 917.567, 'start': 899.521, 'weight': 0.75}], 'summary': 'Covers the diffusion equation and its application in understanding heat flow, along with molecular diffusion, modeling particle movement, flux, density calculation, and the connection to partial derivatives, providing in-depth insights into these concepts and their practical implications.', 'chapters': [{'end': 50.776, 'segs': [{'end': 50.776, 'src': 'embed', 'start': 16.187, 'weight': 0, 'content': [{'end': 18.728, 'text': 'So the diffusion equation describes how things spread.', 'start': 16.187, 'duration': 2.541}, {'end': 25.971, 'text': "For example, in other contexts, it's known as the heat equation, and it can tell you how temperature flows from hot areas to cool areas.", 'start': 19.208, 'duration': 6.763}, {'end': 30.152, 'text': "So if I were to bring together some kind of hot rod, let's say it was 90 degrees.", 'start': 26.451, 'duration': 3.701}, {'end': 31.893, 'text': "with a cooler rod let's say 10 degrees.", 'start': 30.152, 'duration': 1.741}, {'end': 36.759, 'text': 'you can know that the temperature will generally flow from the hot portion to the cool portion.', 'start': 32.393, 'duration': 4.366}, {'end': 40.323, 'text': 'Over time, that left half gets colder and that right half gets warmer.', 'start': 37.119, 'duration': 3.204}, {'end': 42.626, 'text': 'But how specifically does this happen?', 'start': 40.704, 'duration': 1.922}, {'end': 43.968, 'text': 'How long is it gonna take?', 'start': 42.907, 'duration': 1.061}, {'end': 50.776, 'text': "If we take a freeze frame at a given point in time, what's the shape of that temperature distribution across the pair of rods going to be?", 'start': 44.348, 'duration': 6.428}], 'summary': 'Diffusion equation models temperature flow from hot to cool areas, affecting rod temperatures over time.', 'duration': 34.589, 'max_score': 16.187, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c16187.jpg'}], 'start': 0.089, 'title': 'Diffusion equation and heat flow', 'summary': 'Explores the diffusion equation, a partial differential equation describing the spread of substances such as heat, and its application in understanding the flow of temperature from hot to cool areas.', 'chapters': [{'end': 50.776, 'start': 0.089, 'title': 'Diffusion equation and heat flow', 'summary': 'Explores the diffusion equation, a partial differential equation describing the spread of substances such as heat, and its application in understanding the flow of temperature from hot to cool areas.', 'duration': 50.687, 'highlights': ['The diffusion equation, also known as the heat equation, explains the flow of temperature from hot areas to cool areas, providing insights into how temperature distribution changes over time.', 'It demonstrates how temperature flows from a hot rod to a cooler rod, with the left half getting colder and the right half getting warmer over time.', 'The lecture aims to illustrate the origin of the diffusion equation, which plays a crucial role in understanding the spread and distribution of substances, including heat.']}], 'duration': 50.687, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c89.jpg', 'highlights': ['The diffusion equation, also known as the heat equation, explains the flow of temperature from hot areas to cool areas, providing insights into how temperature distribution changes over time.', 'It demonstrates how temperature flows from a hot rod to a cooler rod, with the left half getting colder and the right half getting warmer over time.', 'The lecture aims to illustrate the origin of the diffusion equation, which plays a crucial role in understanding the spread and distribution of substances, including heat.']}, {'end': 325.707, 'segs': [{'end': 77.118, 'src': 'embed', 'start': 51.497, 'weight': 3, 'content': [{'end': 56.922, 'text': "And what's nice is the same equation that describes heat flow like this also describes all sorts of other diffusion.", 'start': 51.497, 'duration': 5.425}, {'end': 61.225, 'text': "So, for example, in this lecture, we're gonna focus on the idea of molecular diffusion,", 'start': 57.322, 'duration': 3.903}, {'end': 65.649, 'text': 'where you might interpret the same graph as instead describing the density of a gas.', 'start': 61.225, 'duration': 4.424}, {'end': 71.374, 'text': 'If you were to have a room where that density was really high in one part and really low in another part.', 'start': 66.049, 'duration': 5.325}, {'end': 77.118, 'text': 'over time, the gas particles will tend to move from the high density part towards the low density part on average.', 'start': 71.374, 'duration': 5.744}], 'summary': 'Heat equation describes diffusion, including molecular diffusion of gas particles.', 'duration': 25.621, 'max_score': 51.497, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c51497.jpg'}, {'end': 138.132, 'src': 'embed', 'start': 108.692, 'weight': 5, 'content': [{'end': 112.114, 'text': 'and then we say ignore the other molecular interactions,', 'start': 108.692, 'duration': 3.422}, {'end': 116.117, 'text': "pretend they do this random walk but then don't account for how they interact with each other.", 'start': 112.114, 'duration': 4.003}, {'end': 118.078, 'text': "that's what's going to get us, this diffusion equation.", 'start': 116.117, 'duration': 1.961}, {'end': 124.402, 'text': "So, in particular, I should emphasize we're not modeling things like pressure where, if you have you know, high density gas,", 'start': 118.658, 'duration': 5.744}, {'end': 132.368, 'text': "part of what's making it move towards the low density region is the increased number of collisions and the increased number of interactions happening in that high density part.", 'start': 124.402, 'duration': 7.966}, {'end': 138.132, 'text': "Here we're just going to describe why that spread happens from the statistics alone of what a random walk would give you.", 'start': 132.788, 'duration': 5.344}], 'summary': 'Describing the diffusion equation, modeling without accounting for molecular interactions or pressure, focusing on statistics of random walk.', 'duration': 29.44, 'max_score': 108.692, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c108692.jpg'}, {'end': 186.617, 'src': 'embed', 'start': 159.798, 'weight': 0, 'content': [{'end': 166.323, 'text': "And then what we'll do is we'll say what happens as this time step gets smaller and smaller and as the spatial step gets smaller and smaller,", 'start': 159.798, 'duration': 6.525}, {'end': 170.065, 'text': 'so that we see how the discrete model gives rise to a continuous model.', 'start': 166.323, 'duration': 3.742}, {'end': 176.89, 'text': 'So just to see what this looks like, let me play out that same animation, but with 100 different dots instead of just the one.', 'start': 171.146, 'duration': 5.744}, {'end': 182.834, 'text': 'So here I have 100 dots sitting in that middle and if I let it play out where, with each time step they randomly.', 'start': 177.51, 'duration': 5.324}, {'end': 186.617, 'text': 'I think in this case they either stay where they are or they take a random step to one of their neighbors.', 'start': 182.834, 'duration': 3.783}], 'summary': 'Model explores transition from discrete to continuous with 100 dots taking random steps.', 'duration': 26.819, 'max_score': 159.798, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c159798.jpg'}, {'end': 231.906, 'src': 'embed', 'start': 209.359, 'weight': 4, 'content': [{'end': 217.402, 'text': 'And what we want to know is how can we describe that spread with a function or an equation that we can solve to get some kind of relevant function.', 'start': 209.359, 'duration': 8.043}, {'end': 220.223, 'text': 'So modeling it with random walks and discretizing space.', 'start': 217.782, 'duration': 2.441}, {'end': 223.464, 'text': 'this is all in the spirit of getting the simplest possible thing we can.', 'start': 220.223, 'duration': 3.241}, {'end': 225.385, 'text': "that's going to give rise to this equation.", 'start': 223.464, 'duration': 1.921}, {'end': 231.906, 'text': "And we're actually going to go one step simpler and say it's easier to think about things if they just move in one dimension rather than two.", 'start': 225.985, 'duration': 5.921}], 'summary': 'Modeling spread with a simple one-dimensional equation.', 'duration': 22.547, 'max_score': 209.359, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c209359.jpg'}, {'end': 288.336, 'src': 'embed', 'start': 262.083, 'weight': 1, 'content': [{'end': 266.068, 'text': "And then I'll let each one of these molecules move according to the rule that I just described.", 'start': 262.083, 'duration': 3.985}, {'end': 270.013, 'text': "And what we're going to see is a diffusion from the dense part to the lesser dense part.", 'start': 266.468, 'duration': 3.545}, {'end': 275.099, 'text': 'So for example, after one time step, about half of the ones on the right edge move to the right.', 'start': 270.353, 'duration': 4.746}, {'end': 282.028, 'text': 'And then after another time step you know, another half of those are, I guess based on the random statistics of here a little more than half moved,', 'start': 275.54, 'duration': 6.488}, {'end': 282.749, 'text': 'one more step there.', 'start': 282.028, 'duration': 0.721}, {'end': 288.336, 'text': 'And all in all, as this thing plays out, we see a general tendency to flow from the left to the right.', 'start': 283.35, 'duration': 4.986}], 'summary': 'Simulation shows diffusion from dense to less dense, with about half moving to the right after each time step.', 'duration': 26.253, 'max_score': 262.083, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c262083.jpg'}], 'start': 51.497, 'title': 'Molecular diffusion and random walk', 'summary': 'Discusses molecular diffusion, where gas particles move from high to low density areas over time, based on the statistics of a random walk. it also explores discrete random walks and how a discrete model transitions to a continuous model with examples of 100 and 10,000 dots, emphasizing the need for separate boundary rules.', 'chapters': [{'end': 138.132, 'start': 51.497, 'title': 'Molecular diffusion and random walk', 'summary': 'Discusses the concept of molecular diffusion, where gas particles tend to move from high density to low density areas over time, based on the statistics of a random walk.', 'duration': 86.635, 'highlights': ['The concept of molecular diffusion is explained, where gas particles move from high density to low density areas over time, based on the statistics of a random walk.', 'The equation describing heat flow also encompasses various diffusion processes, including molecular diffusion.', 'The lecture emphasizes modeling the interaction of molecules as a random walk, disregarding individual molecular interactions, to derive the diffusion equation.']}, {'end': 325.707, 'start': 138.832, 'title': 'Discrete random walk model', 'summary': 'Discusses the concept of discrete random walks, showing how a discrete model gives rise to a continuous model with examples of 100, 10,000 dots, and modeling diffusion from high to low density, emphasizing the need for separate boundary rules.', 'duration': 186.875, 'highlights': ['The discrete model gives rise to a continuous model as the time step and spatial step get smaller, demonstrated with examples of 100 and 10,000 dots spreading across space. The discrete model transforms into a continuous model as the time step and spatial step decrease, illustrated by the spread of 100 and 10,000 dots across space.', 'Modeling diffusion from high to low density using the discrete random walk model, with a demonstration of particles flowing from the dense part to the lesser dense part. Demonstration of modeling diffusion from high to low density using the discrete random walk model, depicting particles flowing from the dense part to the lesser dense part.', 'The need for separate boundary rules in the context of discrete random walks, emphasizing the importance of defining specific rules for what happens at the boundary. Emphasis on the necessity of separate boundary rules in the context of discrete random walks and the importance of defining specific rules for boundary conditions.']}], 'duration': 274.21, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c51497.jpg', 'highlights': ['The discrete model transforms into a continuous model as the time step and spatial step decrease, illustrated by the spread of 100 and 10,000 dots across space.', 'Modeling diffusion from high to low density using the discrete random walk model, with a demonstration of particles flowing from the dense part to the lesser dense part.', 'The concept of molecular diffusion is explained, where gas particles move from high density to low density areas over time, based on the statistics of a random walk.', 'The equation describing heat flow also encompasses various diffusion processes, including molecular diffusion.', 'The need for separate boundary rules in the context of discrete random walks, emphasizing the importance of defining specific rules for what happens at the boundary.', 'The lecture emphasizes modeling the interaction of molecules as a random walk, disregarding individual molecular interactions, to derive the diffusion equation.']}, {'end': 476.328, 'segs': [{'end': 380.09, 'src': 'embed', 'start': 342.016, 'weight': 0, 'content': [{'end': 344.097, 'text': 'you do often have to think specially about the boundary.', 'start': 342.016, 'duration': 2.081}, {'end': 349.981, 'text': "So with all of that as a setup, let's go ahead and pull up some paper and actually describe this a little bit more mathematically.", 'start': 344.998, 'duration': 4.983}, {'end': 356.215, 'text': "So in this setup, we're thinking of space very discretely, almost as if it was divided into these different cells,", 'start': 350.751, 'duration': 5.464}, {'end': 358.296, 'text': 'and each cell has some number of particles in it.', 'start': 356.215, 'duration': 2.081}, {'end': 366.281, 'text': 'And the rule is that at every time step, so after some time step, delta, t half of the particles,', 'start': 358.796, 'duration': 7.485}, {'end': 369.323, 'text': 'or maybe we should say each particle with 50% probability,', 'start': 366.281, 'duration': 3.042}, {'end': 376.468, 'text': 'will move one unit to the left and then each particle with 50% probability will move one unit sorry, to the right,', 'start': 369.323, 'duration': 7.145}, {'end': 380.09, 'text': 'excuse me and then the other half move one unit to the left.', 'start': 376.468, 'duration': 3.622}], 'summary': 'Describing a mathematical model for particle movement in discrete space.', 'duration': 38.074, 'max_score': 342.016, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c342016.jpg'}, {'end': 444.632, 'src': 'embed', 'start': 419.393, 'weight': 1, 'content': [{'end': 424.477, 'text': "you know the densities of all of its neighbors? And this is not too hard to think about, but let's go ahead and write it out.", 'start': 419.393, 'duration': 5.084}, {'end': 425.558, 'text': "It'll be pretty elucidating.", 'start': 424.537, 'duration': 1.021}, {'end': 433.323, 'text': "So, this change, which in the back of our minds we're thinking of as something that happens because of time, as you take a step forward in time.", 'start': 426.238, 'duration': 7.085}, {'end': 439.708, 'text': "Well, let's also say this is a change at a particular index, so maybe we're thinking of rho sub five.", 'start': 433.343, 'duration': 6.365}, {'end': 444.632, 'text': 'Well, the things adding to it are those coming from the right and coming from the left.', 'start': 440.429, 'duration': 4.203}], 'summary': 'Discussion on density change at a specific index due to neighboring values.', 'duration': 25.239, 'max_score': 419.393, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c419393.jpg'}], 'start': 326.107, 'title': 'Modeling particle movement', 'summary': 'Discusses a mathematical model for particle movement in discrete space, where particles move with 50% probability to the left or right, determining the change in density at each time step based on the densities of its neighbors.', 'chapters': [{'end': 476.328, 'start': 326.107, 'title': 'Modeling particle movement in discrete space', 'summary': 'Discusses the mathematical model for particle movement in discrete space, where particles move with 50% probability to the left or right, determining the change in density at each time step based on the densities of its neighbors.', 'duration': 150.221, 'highlights': ['Particles move with 50% probability to the left or right, affecting the density at each time step. The movement of particles in discrete space is governed by a rule where each particle has a 50% chance of moving one unit to the left or right, influencing the density at each time step.', 'Determining the change in density based on the densities of neighboring cells. The change in density at a particular index is influenced by the densities of its neighboring cells, with half of the particles from the right and left affecting the change.', 'Mathematical model for particle movement in discrete space. The discussion presents a mathematical model for describing the movement of particles in discrete space, dividing space into cells and analyzing the change in density at each time step.']}], 'duration': 150.221, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c326107.jpg', 'highlights': ['Particles move with 50% probability to the left or right, affecting the density at each time step.', 'Determining the change in density based on the densities of neighboring cells.', 'Mathematical model for particle movement in discrete space.']}, {'end': 823.865, 'segs': [{'end': 701.97, 'src': 'heatmap', 'start': 505.242, 'weight': 0, 'content': [{'end': 507.184, 'text': 'And you can kind of think of what each of these terms mean.', 'start': 505.242, 'duration': 1.942}, {'end': 513.87, 'text': 'This one here, rho sub n plus one minus rho sub n, is telling us the difference in density across this barrier.', 'start': 507.604, 'duration': 6.266}, {'end': 517.832, 'text': 'That difference is determining the flux of particles between them.', 'start': 514.13, 'duration': 3.702}, {'end': 522.277, 'text': 'And then what we subtract off is the difference in density on the left boundary.', 'start': 518.554, 'duration': 3.723}, {'end': 526.201, 'text': "So it's kind of like we're taking a look at the flux on the right and the flux on the left.", 'start': 522.977, 'duration': 3.224}, {'end': 530.345, 'text': "That's what's determining the net change of particles in this particular cell.", 'start': 526.721, 'duration': 3.624}, {'end': 539.097, 'text': "And because this expression is a kind of difference of differences, I'm going to go ahead and write it with a funny little standard notation here,", 'start': 530.965, 'duration': 8.132}, {'end': 542.822, 'text': "where I'm going to call it delta, delta rho.", 'start': 539.097, 'duration': 3.725}, {'end': 545.322, 'text': 'around n.', 'start': 544.221, 'duration': 1.101}, {'end': 551.725, 'text': 'If you look at the change in density across this right barrier and then look at the change in density across this left barrier,', 'start': 545.322, 'duration': 6.403}, {'end': 554.466, 'text': 'the difference between those two changes is the term that we want.', 'start': 551.725, 'duration': 2.741}, {'end': 561.09, 'text': "But there's an ambiguity in the notation here, because over here I'm describing a change in density with respect to time,", 'start': 555.267, 'duration': 5.823}, {'end': 566.913, 'text': "so maybe I should put a little subscript on that delta, but here we're describing changes in density with respect to space.", 'start': 561.09, 'duration': 5.823}, {'end': 570.655, 'text': "So I'll put a little subscript x there for those.", 'start': 567.313, 'duration': 3.342}, {'end': 582.352, 'text': 'So this right here is the core idea that the change in density for each unit time for each time step is proportional to this kind of second order change as we move spatially.', 'start': 571.376, 'duration': 10.976}, {'end': 584.32, 'text': 'Now. ultimately,', 'start': 583.72, 'duration': 0.6}, {'end': 591.762, 'text': 'we want to consider what happens as this tiny time step approaches zero and as the steps in our discrete units of space also approach zero.', 'start': 584.32, 'duration': 7.442}, {'end': 595.883, 'text': 'Which is to say we want to understand what does this rule look like in a continuous context.', 'start': 591.922, 'duration': 3.961}, {'end': 599.444, 'text': 'But for that we have to start talking about rates instead of absolute differences.', 'start': 596.303, 'duration': 3.141}, {'end': 607.585, 'text': 'So what we really want to look at is something like the change over time for this density divided by the size of that time step.', 'start': 599.804, 'duration': 7.781}, {'end': 610.206, 'text': 'That way we can take its limit and get some kind of derivative.', 'start': 607.645, 'duration': 2.561}, {'end': 612.727, 'text': 'And then similarly on the other side.', 'start': 610.746, 'duration': 1.981}, {'end': 619.352, 'text': 'what we want to see is not just this second order change of delta, delta rho,', 'start': 612.727, 'duration': 6.625}, {'end': 625.036, 'text': 'but we want to know that second order change divided by the spatial step.', 'start': 619.352, 'duration': 5.684}, {'end': 631.44, 'text': 'But it should properly be the size of that spatial step squared, because if we take a second order difference like this,', 'start': 625.156, 'duration': 6.284}, {'end': 634.762, 'text': 'then as we shrink down our spatial steps,', 'start': 631.44, 'duration': 3.322}, {'end': 640.366, 'text': 'the size of the second order change will be roughly proportional to the square of the size of that difference.', 'start': 634.762, 'duration': 5.604}, {'end': 649.831, 'text': 'One way you might think about this is if you were to take, you know, just the change across space of rho, divided by the size of those spatial steps.', 'start': 641.346, 'duration': 8.485}, {'end': 652.692, 'text': 'this will approach a derivative of density with respect to space.', 'start': 649.831, 'duration': 2.861}, {'end': 658.675, 'text': 'And then if we did the same thing to that term, we said hey, how much do you change as we take little steps in space,', 'start': 653.192, 'duration': 5.483}, {'end': 660.696, 'text': 'divided by the size of those steps?', 'start': 658.675, 'duration': 2.021}, {'end': 664.038, 'text': 'This whole thing will now approach a kind of second derivative.', 'start': 661.857, 'duration': 2.181}, {'end': 671.589, 'text': "But, of course, if we're going to be adding these new terms, like the size of our time step or the size of our spatial step to the equation,", 'start': 665.506, 'duration': 6.083}, {'end': 672.669, 'text': "we've got to balance things out.", 'start': 671.589, 'duration': 1.08}, {'end': 679.632, 'text': 'So we might say this whole thing should be multiplied by the size of delta x squared all divided by delta t.', 'start': 672.729, 'duration': 6.903}, {'end': 683.974, 'text': "But let's go ahead and just wrap up all of this and also that 1 half as a constant.", 'start': 679.632, 'duration': 4.342}, {'end': 690.598, 'text': "So if we've got our change in density with respect to time.", 'start': 684.394, 'duration': 6.204}, {'end': 701.97, 'text': "This is going to equal some constant that I'll call d times this second order change with respect to space.", 'start': 690.618, 'duration': 11.352}], 'summary': 'The change in density over time is proportional to the second order change with respect to space.', 'duration': 40.08, 'max_score': 505.242, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c505242.jpg'}, {'end': 732.103, 'src': 'heatmap', 'start': 708.359, 'weight': 0.703, 'content': [{'end': 716.268, 'text': "it'll be equal to one half times the step size for our particle squared, divided by the time steps that we have.", 'start': 708.359, 'duration': 7.909}, {'end': 723.035, 'text': 'So if our particles were taking larger steps with each one of these rules, then this diffusivity constant would be larger.', 'start': 717.169, 'duration': 5.866}, {'end': 724.677, 'text': 'It would actually grow quadratically with that.', 'start': 723.075, 'duration': 1.602}, {'end': 732.103, 'text': 'And similarly, if we had smaller and smaller time steps, meaning the amount of time that it takes for each one of these jumps to happen was smaller,', 'start': 725.237, 'duration': 6.866}], 'summary': 'Diffusivity constant increases quadratically with larger particle steps and decreases with smaller time steps.', 'duration': 23.744, 'max_score': 708.359, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c708359.jpg'}, {'end': 769.673, 'src': 'embed', 'start': 742.471, 'weight': 3, 'content': [{'end': 750.922, 'text': 'now we can say as we consider smaller and smaller time steps, so delta t goes to zero and smaller and smaller spatial steps, delta x goes to zero,', 'start': 742.471, 'duration': 8.451}, {'end': 754.729, 'text': 'but we do it in such a way that this term d stays constant.', 'start': 750.922, 'duration': 3.807}, {'end': 758.497, 'text': "so there's a relationship to how our time steps and our spatial steps are going to zero.", 'start': 754.729, 'duration': 3.768}, {'end': 763.431, 'text': 'Then in the limit, we can think of these as derivatives of a continuous density function.', 'start': 759.11, 'duration': 4.321}, {'end': 769.673, 'text': 'So on the left-hand side, we would have the partial derivative of density with respect to t.', 'start': 763.811, 'duration': 5.862}], 'summary': 'As time and spatial steps approach zero, we have a constant term with a relationship between them, leading to derivatives of a continuous density function.', 'duration': 27.202, 'max_score': 742.471, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c742471.jpg'}, {'end': 807.902, 'src': 'embed', 'start': 781.917, 'weight': 2, 'content': [{'end': 791.071, 'text': 'And this is going to be proportional with this proportionality constant of diffusivity to the second partial derivative of our density with respect to space.', 'start': 781.917, 'duration': 9.154}, {'end': 795.975, 'text': "So this expression is what's known as the diffusion equation.", 'start': 793.294, 'duration': 2.681}, {'end': 802.099, 'text': 'And despite having a fair bit of fancy symbols in it, the idea is that it can arise from a very, very simple context.', 'start': 796.316, 'duration': 5.783}, {'end': 807.902, 'text': 'In this case, just a random walk model with particles stepping either to the left or to the right with 50% probability.', 'start': 802.399, 'duration': 5.503}], 'summary': 'Diffusion equation arises from simple random walk model with 50% probability.', 'duration': 25.985, 'max_score': 781.917, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c781917.jpg'}], 'start': 476.468, 'title': 'Flux, density, and diffusion equation', 'summary': 'Explains flux and density calculation, highlighting the net change of particles in a cell and the ambiguity in notation, and delves into diffusivity, diffusion equation, and their relationship with time and spatial steps in a continuous context.', 'chapters': [{'end': 561.09, 'start': 476.468, 'title': 'Flux and density calculation', 'summary': 'Explains the calculation of flux and density differences in a physical system, highlighting the net change of particles in a cell, determined by the difference in density across barriers and the ambiguity in the notation regarding change in density with respect to time.', 'duration': 84.622, 'highlights': ['The difference in density across a barrier (rho sub n plus one minus rho sub n) determines the flux of particles between them.', 'The subtraction of the difference in density on the left boundary (rho sub n minus row sub n minus one) helps determine the net change of particles in a cell.', 'The expression delta, delta rho around n represents the difference between the changes in density across the right and left barriers.', "There's an ambiguity in the notation regarding the change in density with respect to time."]}, {'end': 823.865, 'start': 561.09, 'title': 'Diffusion equation and continuous context', 'summary': 'Explains the concept of diffusivity and the diffusion equation, emphasizing the relationship between time steps, spatial steps, and diffusivity, with a clear definition and application in a continuous context.', 'duration': 262.775, 'highlights': ['The concept of diffusivity is explained as a proportionality constant that represents the rate of change in density with respect to space and time steps, with a clear definition and example of its variation based on time steps and spatial steps. Diffusivity constant defined as one half times the step size for particles squared, divided by the time steps.', "The relationship between time steps, spatial steps, and diffusivity is emphasized, with the constant 'd' remaining constant as time steps and spatial steps approach zero, leading to the consideration of derivatives of a continuous density function. The relationship between time steps, spatial steps, and diffusivity explained as time steps and spatial steps approach zero, 'd' remains constant, leading to derivatives of a continuous density function.", 'The diffusion equation is introduced as a result of these concepts, with a clear explanation of its components and its simple origin from a random walk model, emphasizing its readability in a continuous function context. Introduction of the diffusion equation, emphasizing its simple origin from a random walk model and readability in a continuous function context.']}], 'duration': 347.397, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c476468.jpg', 'highlights': ['The difference in density across a barrier determines the flux of particles between them.', 'The subtraction of the difference in density helps determine the net change of particles in a cell.', 'The concept of diffusivity is explained as a proportionality constant representing the rate of change in density with respect to space and time steps.', "The relationship between time steps, spatial steps, and diffusivity is emphasized, with the constant 'd' remaining constant as time steps and spatial steps approach zero.", 'The diffusion equation is introduced as a result of these concepts, with a clear explanation of its components and its simple origin from a random walk model.']}, {'end': 958.835, 'segs': [{'end': 921.369, 'src': 'heatmap', 'start': 886.766, 'weight': 0, 'content': [{'end': 893.991, 'text': "It says the way that things change in one dimension is inextricably linked with the way that it's changing in another direction.", 'start': 886.766, 'duration': 7.225}, {'end': 899.061, 'text': "So in particular, let's go ahead and kind of read off what it is that our diffusion equation is telling us.", 'start': 894.639, 'duration': 4.422}, {'end': 905.663, 'text': 'The second partial derivative term with respect to space describes the curvature of this function.', 'start': 899.521, 'duration': 6.142}, {'end': 912.765, 'text': "Points where that second derivative are negative tend to be curving down, meaning they're points where the neighbors are lower than what they are.", 'start': 906.083, 'duration': 6.682}, {'end': 917.567, 'text': 'So it makes sense that the molecules are going to tend to diffuse away, or the heat will diffuse away.', 'start': 912.785, 'duration': 4.782}, {'end': 921.369, 'text': 'Points where that second derivative are higher are points where this curves up.', 'start': 918.187, 'duration': 3.182}], 'summary': 'Diffusion equation describes how molecules or heat diffuse based on curvature of function.', 'duration': 34.603, 'max_score': 886.766, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c886766.jpg'}, {'end': 970.386, 'src': 'embed', 'start': 939.023, 'weight': 1, 'content': [{'end': 942.227, 'text': 'especially sharp curves, those are the ones that change the most quickly.', 'start': 939.023, 'duration': 3.204}, {'end': 949.535, 'text': 'So over time, the heat distribution or the molecular distribution, whatever it might be, tends to get smoother over time.', 'start': 942.647, 'duration': 6.888}, {'end': 951.296, 'text': "That's a consequence of this equation.", 'start': 949.575, 'duration': 1.721}, {'end': 956.633, 'text': 'Now you might wonder how this whole setup changes if, instead of thinking of one-dimensional motion,', 'start': 952.369, 'duration': 4.264}, {'end': 958.835, 'text': 'we start thinking of two- or three-dimensional motion.', 'start': 956.633, 'duration': 2.202}, {'end': 961.918, 'text': 'And the nice part is, it actually hardly changes at all.', 'start': 959.356, 'duration': 2.562}, {'end': 970.386, 'text': 'So if our cell, instead of having neighbors just to the left and to the right, also had some neighbors to the top and the bottom?', 'start': 962.559, 'duration': 7.827}], 'summary': 'Heat distribution smooths over time, equation affects motion in 2d/3d minimally.', 'duration': 31.363, 'max_score': 939.023, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c939023.jpg'}], 'start': 823.865, 'title': 'Diffusion equation and partial derivatives', 'summary': "Explores the diffusion equation's connection to temperature and molecular diffusion, highlighting the role of partial derivatives and the evolution towards a more uniform distribution over time.", 'chapters': [{'end': 958.835, 'start': 823.865, 'title': 'Diffusion equation and partial derivatives', 'summary': 'Explains how the diffusion equation relates to temperature diffusion and molecular diffusion, emphasizing the concept of partial derivatives and the relationship between changes in space and time, ultimately leading to a smoother distribution over time.', 'duration': 134.97, 'highlights': ['The diffusion equation relates to temperature diffusion and molecular diffusion, emphasizing the concept of partial derivatives and the relationship between changes in space and time. The same equation describes molecular diffusion and temperature diffusion, utilizing partial derivatives to show the relationship between changes in space and time.', 'The relationship between changes in space and time leads to a smoother distribution over time. The equation shows that the way each point changes in time is proportional to the curvature at that point, leading to a smoother heat or molecular distribution over time.', 'The second partial derivative term with respect to space describes the curvature of the function, influencing the tendency for diffusion or heat to spread. The second partial derivative term with respect to space describes the curvature of the function, determining the tendency for diffusion or heat to spread based on the curvature of the function.']}], 'duration': 134.97, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c823865.jpg', 'highlights': ['The diffusion equation relates to temperature and molecular diffusion, emphasizing partial derivatives and the relationship between changes in space and time.', 'The equation shows that the way each point changes in time is proportional to the curvature at that point, leading to a smoother heat or molecular distribution over time.', 'The second partial derivative term with respect to space describes the curvature of the function, influencing the tendency for diffusion or heat to spread.']}, {'end': 1288.239, 'segs': [{'end': 1064.379, 'src': 'embed', 'start': 1039.123, 'weight': 1, 'content': [{'end': 1047.229, 'text': 'And then very similarly, if we were doing this in three dimensions, you would also have to consider the change in flux across the boundaries,', 'start': 1039.123, 'duration': 8.106}, {'end': 1049.33, 'text': 'kind of in and out of space as you go.', 'start': 1047.229, 'duration': 2.101}, {'end': 1057.395, 'text': "And this idea here of adding together a bunch of second derivative terms actually comes up commonly enough that there's compact notation for it.", 'start': 1050.291, 'duration': 7.104}, {'end': 1064.379, 'text': 'We write it as a upside down triangle squared of our density function, rho.', 'start': 1058.056, 'duration': 6.323}], 'summary': 'Discusses change in flux across boundaries in three dimensions and compact notation for second derivative terms.', 'duration': 25.256, 'max_score': 1039.123, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c1039123.jpg'}, {'end': 1140.496, 'src': 'embed', 'start': 1114.68, 'weight': 0, 'content': [{'end': 1120.063, 'text': "This operator where we're adding up all of the different second partial derivative terms is known as the Laplacian.", 'start': 1114.68, 'duration': 5.383}, {'end': 1123.585, 'text': "And here's one way that you can think about it in terms of the graph of a function.", 'start': 1120.543, 'duration': 3.042}, {'end': 1129.408, 'text': "So here I've got a function where the input is two-dimensional, it's got x and y coordinates and then the output is just a number.", 'start': 1124.045, 'duration': 5.363}, {'end': 1130.408, 'text': "so we're graphing it above.", 'start': 1129.408, 'duration': 1}, {'end': 1136.412, 'text': 'And one way to think about this Laplacian is to say, look at all of the neighboring points around a given point.', 'start': 1130.949, 'duration': 5.463}, {'end': 1140.496, 'text': 'consider the value of the function at those neighboring points,', 'start': 1137.032, 'duration': 3.464}], 'summary': 'The laplacian operator computes second partial derivatives in two-dimensional space.', 'duration': 25.816, 'max_score': 1114.68, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c1114680.jpg'}, {'end': 1199.287, 'src': 'embed', 'start': 1168.59, 'weight': 2, 'content': [{'end': 1172.133, 'text': 'Okay, so there are a couple last things that I want to bring up before wrapping things up here.', 'start': 1168.59, 'duration': 3.543}, {'end': 1173.254, 'text': 'First of all,', 'start': 1172.653, 'duration': 0.601}, {'end': 1185.104, 'text': "this effect that I've described of the molecular diffusion that's based just on the random walks of the molecules themselves and the statistics at play there is actually a relatively small effect when it comes to looking at the overall diffusion in a gas.", 'start': 1173.254, 'duration': 11.85}, {'end': 1190.705, 'text': "Most of it ends up coming from larger scale phenomena, like the turbulent flows that you'll get.", 'start': 1185.664, 'duration': 5.041}, {'end': 1199.287, 'text': "And that's the kind of thing that arises once you start considering how these molecules interact with each other and all of the really hairy nonlinear complexities that are involved in that.", 'start': 1191.105, 'duration': 8.182}], 'summary': 'Molecular diffusion has a relatively small effect on gas diffusion compared to turbulent flows and molecule interactions.', 'duration': 30.697, 'max_score': 1168.59, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c1168590.jpg'}, {'end': 1225.704, 'src': 'embed', 'start': 1200.007, 'weight': 5, 'content': [{'end': 1207.149, 'text': 'But the main thing I actually want to highlight is this kind of interesting dual nature to the interplay between discrete models and continuous models.', 'start': 1200.007, 'duration': 7.142}, {'end': 1213.714, 'text': 'So here I showed how we can get this continuous model for diffusion that arises from a really simple discrete setup.', 'start': 1207.669, 'duration': 6.045}, {'end': 1220.58, 'text': "But what's funny is then, when we go and we actually use this diffusion equation to model things like climate or whatever else,", 'start': 1214.234, 'duration': 6.346}, {'end': 1225.704, 'text': 'we find ourselves using computers to discretize what was originally described continuously.', 'start': 1220.58, 'duration': 5.124}], 'summary': 'Interplay between discrete and continuous models is highlighted, showing the use of computers to discretize continuous models.', 'duration': 25.697, 'max_score': 1200.007, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c1200007.jpg'}, {'end': 1265.187, 'src': 'embed', 'start': 1241.011, 'weight': 3, 'content': [{'end': 1247.574, 'text': 'So what you do is you get the computer to discretize space and to discretize time and just take different steps,', 'start': 1241.011, 'duration': 6.563}, {'end': 1251.116, 'text': 'changing the function according to the rule described by the differential equation.', 'start': 1247.574, 'duration': 3.542}, {'end': 1259.502, 'text': 'which leaves us with this kind of funny situation where we have something that we took pains to take the discrete setup and then turn it into a continuous one and then,', 'start': 1251.616, 'duration': 7.886}, {'end': 1265.187, 'text': 'by the time we actually start doing something meaningful and physical with it, we just turn it right back into a discrete one.', 'start': 1259.502, 'duration': 5.685}], 'summary': 'Using computer to discretize space and time, changing function according to differential equation.', 'duration': 24.176, 'max_score': 1241.011, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c1241011.jpg'}], 'start': 959.356, 'title': 'Analyzing differential equations', 'summary': 'Covers partial derivatives, laplacian operator, and the interplay between discrete and continuous models in differential equations, essential for computational handling.', 'chapters': [{'end': 1199.287, 'start': 959.356, 'title': 'Partial derivatives and laplacian operator', 'summary': 'Discusses the concept of partial derivatives and the laplacian operator, which are used to analyze changes in flux and density in multiple dimensions, as well as the impact of molecular diffusion on overall gas diffusion.', 'duration': 239.931, 'highlights': ['The Laplacian operator is used to analyze changes in density in multiple dimensions and is represented by the sum of second partial derivatives. The Laplacian operator, represented by the sum of second partial derivatives, is used to analyze changes in density in multiple dimensions, providing insights into how different a given point in space is from its neighbors.', 'Partial derivatives are utilized to analyze changes in flux and density across different dimensions, with a focus on the x and y directions. Partial derivatives are used to analyze changes in flux and density across different dimensions, including the x and y directions, providing a way to understand the net flux of particles from neighboring cells.', 'Molecular diffusion has a relatively small effect on overall gas diffusion, with larger scale phenomena such as turbulent flows playing a more significant role. Molecular diffusion, based on the random walks of molecules, has a relatively small effect on overall gas diffusion, with larger scale phenomena like turbulent flows being more influential.']}, {'end': 1288.239, 'start': 1200.007, 'title': 'Interplay between discrete and continuous models', 'summary': 'Discusses the dual nature of the interplay between discrete and continuous models in differential equations, highlighting the process of converting from discrete to continuous and back to discrete, which is essential for computational handling of differential equations.', 'duration': 88.232, 'highlights': ['The process of converting from discrete to continuous and back to discrete is essential for computational handling of differential equations, as most differential equations, especially partial differential equations, cannot be solved directly.', 'Using computers to discretize what was originally described continuously is a common practice, which involves discretizing space and time and changing the function according to the rule described by the differential equation.', 'The lecture demonstrates how the continuous model for diffusion, arising from a simple discrete setup, is ultimately converted back into a discrete one for practical computational handling.']}], 'duration': 328.883, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/a3V0BJLIo_c/pics/a3V0BJLIo_c959356.jpg', 'highlights': ['The Laplacian operator is used to analyze changes in density in multiple dimensions and is represented by the sum of second partial derivatives.', 'Partial derivatives are utilized to analyze changes in flux and density across different dimensions, with a focus on the x and y directions.', 'Molecular diffusion has a relatively small effect on overall gas diffusion, with larger scale phenomena such as turbulent flows playing a more significant role.', 'The process of converting from discrete to continuous and back to discrete is essential for computational handling of differential equations.', 'Using computers to discretize what was originally described continuously is a common practice, which involves discretizing space and time and changing the function according to the rule described by the differential equation.', 'The lecture demonstrates how the continuous model for diffusion, arising from a simple discrete setup, is ultimately converted back into a discrete one for practical computational handling.']}], 'highlights': ['The diffusion equation, also known as the heat equation, explains the flow of temperature from hot areas to cool areas, providing insights into how temperature distribution changes over time.', 'The discrete model transforms into a continuous model as the time step and spatial step decrease, illustrated by the spread of 100 and 10,000 dots across space.', 'Particles move with 50% probability to the left or right, affecting the density at each time step.', 'The difference in density across a barrier determines the flux of particles between them.', 'The Laplacian operator is used to analyze changes in density in multiple dimensions and is represented by the sum of second partial derivatives.', 'The equation shows that the way each point changes in time is proportional to the curvature at that point, leading to a smoother heat or molecular distribution over time.']}