WEBVTT 00:00:00.000 --> 00:00:02.960 OK, so I want you to picture something for me 00:00:02.960 --> 00:00:05.559 before we get started today. You are back in 00:00:05.559 --> 00:00:08.199 your high school calculus class. Oh, boy. Right. 00:00:08.339 --> 00:00:11.359 Maybe you loved it. Maybe you completely hated 00:00:11.359 --> 00:00:13.679 it. You definitely remember the vibe. You've 00:00:13.679 --> 00:00:15.740 got a piece of graph paper, a really sharp pencil, 00:00:15.740 --> 00:00:20.300 and you draw a curve. And it's smooth. clean. 00:00:20.780 --> 00:00:23.519 You can take a ruler, slide it right along that 00:00:23.519 --> 00:00:26.420 curve, and at any single point, you can draw 00:00:26.420 --> 00:00:28.379 a straight line that perfectly touches it. You 00:00:28.379 --> 00:00:30.379 can find the slope. Right, the tangent line. 00:00:30.679 --> 00:00:33.439 Exactly, the tangent line. And that right there 00:00:33.439 --> 00:00:36.219 is the world, according to Isaac Newton. It's 00:00:36.219 --> 00:00:40.039 this world of sliding down perfectly frictionless 00:00:40.039 --> 00:00:43.740 slopes or planets orbiting in flawless ellipses, 00:00:43.960 --> 00:00:46.460 apples falling in a straight line. It's elegant, 00:00:46.619 --> 00:00:48.539 it is incredibly predictable, and above all, 00:00:48.579 --> 00:00:51.380 it's smooth. It is. It's a really beautiful world. 00:00:51.520 --> 00:00:53.920 I mean, it's the mathematics of the ideal. But 00:00:53.920 --> 00:00:56.979 that actually brings us to the problem and really 00:00:56.979 --> 00:00:59.219 the whole heart of our deep dive today. Because 00:00:59.219 --> 00:01:02.439 that world, it's a lie. A lie. I mean, that's 00:01:02.439 --> 00:01:05.200 a bit harsh for Newton, isn't it? Well, OK. Maybe 00:01:05.200 --> 00:01:07.659 a useful fiction is a better way to put it. Because 00:01:07.659 --> 00:01:09.640 the real world, the one you and I actually live 00:01:09.640 --> 00:01:12.959 in, it isn't smooth. It's rough. It's porous, 00:01:13.180 --> 00:01:15.980 it's jagged. And when you try to take that beautiful 00:01:15.980 --> 00:01:18.700 smooth Newtonian calculus and apply it to something 00:01:18.700 --> 00:01:21.590 messy, like. like water seeping through a cracked 00:01:21.590 --> 00:01:24.390 rock, or the chaotic turbulence inside a jet 00:01:24.390 --> 00:01:26.629 engine. Or even the stock market, right? Exactly. 00:01:26.989 --> 00:01:28.890 The map doesn't just get difficult in those cases, 00:01:29.069 --> 00:01:31.829 it fundamentally breaks. It just completely fails. 00:01:32.310 --> 00:01:35.030 And that is exactly where today's topic comes 00:01:35.030 --> 00:01:38.510 in. We are diving deep into a fascinating stack 00:01:38.510 --> 00:01:41.329 of research on something called the fractal derivative. 00:01:42.250 --> 00:01:44.269 Sometimes you'll see it called the Hausdorff 00:01:44.269 --> 00:01:47.879 derivative in the literature. And honestly, looking 00:01:47.879 --> 00:01:50.219 at these sources today, papers from researchers 00:01:50.219 --> 00:01:53.680 like Wen Chen and Abdana Tangana, it really feels 00:01:53.680 --> 00:01:55.760 like we're rewriting the rule book of physics 00:01:55.760 --> 00:01:58.500 from the ground up. In many ways we really are. 00:01:58.719 --> 00:02:01.200 We're looking at a completely new mathematical 00:02:01.200 --> 00:02:04.319 framework, one that's designed to measure change 00:02:04.319 --> 00:02:07.000 when the environment itself is a fractal. So 00:02:07.000 --> 00:02:09.740 we're going to be getting into anomalous diffusion. 00:02:10.349 --> 00:02:13.189 rethinking the very definition of velocity and 00:02:13.189 --> 00:02:16.310 looking at some wild cutting edge work in what's 00:02:16.310 --> 00:02:19.349 called fractal fractional calculus. So, yeah, 00:02:19.569 --> 00:02:21.889 if you've ever wondered how scientists actually 00:02:21.889 --> 00:02:24.810 model things that just refuse to follow the straight 00:02:24.810 --> 00:02:27.310 and narrow path, things that get stuck, bounce 00:02:27.310 --> 00:02:30.050 around, behave chaotically, this is the deep 00:02:30.050 --> 00:02:32.189 dive for you. Exactly. We're basically asking 00:02:32.189 --> 00:02:35.530 how do we measure speed? when the road is completely 00:02:35.530 --> 00:02:38.050 full of potholes. I love that analogy. Let's 00:02:38.050 --> 00:02:40.310 start right there with the failure of standard 00:02:40.310 --> 00:02:42.189 physics, because I think that's the absolute 00:02:42.189 --> 00:02:44.710 best hook for this. You mentioned that the old 00:02:44.710 --> 00:02:47.689 rules break. Why? What is so inherently special 00:02:47.689 --> 00:02:50.229 about a porous media or an aquifer that just 00:02:50.229 --> 00:02:52.330 makes Isaac Newton's math give up and go home? 00:02:52.550 --> 00:02:54.689 It really comes down to the assumptions we make 00:02:54.689 --> 00:02:57.349 about space itself. Think about classical diffusion. 00:02:57.889 --> 00:03:00.469 This is based on fundamental things like Fick's 00:03:00.469 --> 00:03:03.430 laws or Darcy's law or Fourier's law for heat. 00:03:03.629 --> 00:03:06.370 The classics. Right. And all of these laws are 00:03:06.370 --> 00:03:08.990 built on the assumption of a random walk in free 00:03:08.990 --> 00:03:12.330 space. OK, the random walk. I have heard this 00:03:12.330 --> 00:03:14.469 term. It's usually described as a drunk person 00:03:14.469 --> 00:03:16.990 stumbling around. Yeah, that is the classic analogy 00:03:16.990 --> 00:03:20.849 in physics. Imagine a drunk person in a massive, 00:03:21.229 --> 00:03:23.830 completely empty parking lot. OK. They take a 00:03:23.830 --> 00:03:26.610 step, stumble in a totally random direction, 00:03:27.169 --> 00:03:30.430 take another step. Over time, even though their 00:03:30.430 --> 00:03:33.389 specific path is random, You can statistically 00:03:33.389 --> 00:03:36.090 predict exactly how far they'll be from the center. 00:03:36.830 --> 00:03:40.370 It follows a nice predictable bell curve, a standard 00:03:40.370 --> 00:03:42.610 Gaussian distribution. Right, because the parking 00:03:42.610 --> 00:03:45.129 lot is empty. There is absolutely nothing stopping 00:03:45.129 --> 00:03:47.689 them. The space is smooth. Precisely. But now 00:03:47.689 --> 00:03:50.389 takes that exact same drunk person and drop them 00:03:50.389 --> 00:03:53.409 into a hedge maze. Oh wow. Or a jagged underground 00:03:53.409 --> 00:03:56.050 cave system. or a sponge. Okay, now they're just 00:03:56.050 --> 00:03:58.030 slamming into walls. They're hitting dead ends. 00:03:58.409 --> 00:04:00.129 They're getting trapped in little feedback loops. 00:04:00.370 --> 00:04:02.550 They have to retrace their steps. In the literature, 00:04:02.610 --> 00:04:04.370 this is actually often called the ant in the 00:04:04.370 --> 00:04:06.550 labyrinth problem. The ant in the labyrinth. 00:04:06.569 --> 00:04:08.930 I really like that. Yeah, so if you're an ant 00:04:08.930 --> 00:04:12.330 inside a complex porous rock, the actual distance 00:04:12.330 --> 00:04:15.349 you travel isn't a straight line. It's this jagged 00:04:15.349 --> 00:04:18.170 self -similar fractal path. you might have to 00:04:18.170 --> 00:04:21.389 walk 10 meters just to move one meter of actual 00:04:21.389 --> 00:04:23.850 displacement. Right. So if you try to use standard 00:04:23.850 --> 00:04:26.689 calculus, which just implicitly assumes smooth 00:04:26.689 --> 00:04:28.990 space and straight lines, your predictions will 00:04:28.990 --> 00:04:31.569 be completely, laughably wrong. Because the speedometer 00:04:31.569 --> 00:04:33.449 is basically broken since the road itself is 00:04:33.449 --> 00:04:35.509 broken? That's a perfect way to put it. To fix 00:04:35.509 --> 00:04:37.490 this, mathematicians eventually realized they 00:04:37.490 --> 00:04:39.360 couldn't just... They couldn't just tack on a 00:04:39.360 --> 00:04:41.420 correction factor to the old equations. They 00:04:41.420 --> 00:04:45.160 had to redefine the very core concepts of distance 00:04:45.160 --> 00:04:47.680 and velocity. They had to transform the actual 00:04:47.680 --> 00:04:50.300 scales of space and time. And this is where it 00:04:50.300 --> 00:04:52.199 gets really mind -bending to me. In the source 00:04:52.199 --> 00:04:54.639 material, it talks about transforming these scales. 00:04:55.240 --> 00:04:56.920 Instead of just looking at standard position 00:04:56.920 --> 00:04:59.579 x and standard time t, we are suddenly looking 00:04:59.579 --> 00:05:03.149 at x to the power of beta. and t to the power 00:05:03.149 --> 00:05:06.170 of alpha. Yes. This is the fundamental ground 00:05:06.170 --> 00:05:08.569 -up shift. We are officially entering fractal 00:05:08.569 --> 00:05:12.050 spacetime. In the math, this is denoted as S 00:05:12.050 --> 00:05:15.810 super alpha comma beta. S alpha. Right. And in 00:05:15.810 --> 00:05:18.490 this specific space, the traditional definition 00:05:18.490 --> 00:05:21.329 of velocity, which is just change in distance 00:05:21.329 --> 00:05:24.449 divided by change in time, it makes absolutely 00:05:24.449 --> 00:05:26.730 no sense because the space is non -differentiable. 00:05:26.970 --> 00:05:28.670 Non -differentiable. Let me make sure I have 00:05:28.670 --> 00:05:31.560 that. That basically means you cannot draw a 00:05:31.560 --> 00:05:34.259 tangent line, right? Because no matter how far 00:05:34.259 --> 00:05:37.220 you zoom in, it's still jagged. Exactly. Think 00:05:37.220 --> 00:05:39.439 of a classic fractal like the coastline of Britain. 00:05:39.860 --> 00:05:42.079 From space, it looks kind of curvy and smooth. 00:05:42.180 --> 00:05:45.439 You zoom in and it's jagged rocks. Right. You 00:05:45.439 --> 00:05:47.180 zoom in on a single rock. It's full of jagged 00:05:47.180 --> 00:05:49.339 little pebbles and cracks. There is no magical 00:05:49.339 --> 00:05:51.300 point where it suddenly becomes a perfectly smooth 00:05:51.300 --> 00:05:54.180 line. So you literally cannot measure a traditional 00:05:54.180 --> 00:05:56.870 slope. So how on earth do you measure speed in 00:05:56.870 --> 00:05:59.110 a world where there are no smooth roads at all? 00:05:59.230 --> 00:06:02.069 You have to redefine it entirely. In this new 00:06:02.069 --> 00:06:04.670 fractal framework, velocity, let's just call 00:06:04.670 --> 00:06:07.250 it v prime, is defined as the derivative of x 00:06:07.250 --> 00:06:09.250 to the power of beta with respect to t to the 00:06:09.250 --> 00:06:11.370 power of alpha. So dx to the beta divided by 00:06:11.370 --> 00:06:14.189 dt to the alpha. That is just, it's so strange 00:06:14.189 --> 00:06:16.610 to think about. You are not moving through standard 00:06:16.610 --> 00:06:18.829 time anymore. You are moving through fractal 00:06:18.829 --> 00:06:21.529 time. Essentially, yes. You're scaling the time 00:06:21.529 --> 00:06:25.240 and the space to perfectly match the roughness 00:06:25.240 --> 00:06:27.360 of the specific medium you're traveling through. 00:06:27.819 --> 00:06:29.579 I like to think of it like currency exchange 00:06:29.579 --> 00:06:32.420 rates. OK. How so? If you're traveling in a country 00:06:32.420 --> 00:06:35.019 where the local currency is roughness, you can't 00:06:35.019 --> 00:06:37.360 pay for things with standard smooth Newtonian 00:06:37.360 --> 00:06:39.959 dollars. You have to convert them. Oh, I see. 00:06:39.980 --> 00:06:42.639 T to the alpha and x to the beta. Those are your 00:06:42.639 --> 00:06:44.759 converted fractal currency. OK, that makes a 00:06:44.759 --> 00:06:47.680 lot of sense. We are fundamentally calibrating 00:06:47.680 --> 00:06:50.079 our clock and our ruler to match the complexity 00:06:50.079 --> 00:06:52.810 of the maze, but We definitely need to talk about 00:06:52.810 --> 00:06:55.170 the derivative itself. I mean, the outline for 00:06:55.170 --> 00:06:57.449 this deep dive is literally focused on measuring 00:06:57.449 --> 00:06:59.910 the roughness, and the tool for that is the fractal 00:06:59.910 --> 00:07:02.189 derivative. How do we actually calculate this 00:07:02.189 --> 00:07:04.050 thing? I'm looking at the core definition here 00:07:04.050 --> 00:07:06.490 in the papers, and it looks vaguely familiar 00:07:06.490 --> 00:07:08.889 to what I learned in high school, but twisted. 00:07:09.230 --> 00:07:11.670 It is familiar. That's honestly the beauty of 00:07:11.670 --> 00:07:14.230 how they constructed it. In standard calculus, 00:07:14.730 --> 00:07:17.459 the derivative is just a limit. It's the rise 00:07:17.459 --> 00:07:20.439 over run as the run gets infinitely microscopically 00:07:20.439 --> 00:07:24.560 small. You take f of t, 1 minus f of t, and you 00:07:24.560 --> 00:07:26.800 divide it by t1 minus t. Right, just finding 00:07:26.800 --> 00:07:30.720 the slope. Exactly. The fractal derivative rigorously 00:07:30.720 --> 00:07:33.120 keeps that same basic structure, but it swaps 00:07:33.120 --> 00:07:35.420 out the denominator. Instead of measuring the 00:07:35.420 --> 00:07:38.060 change against linear time, which is that t1 00:07:38.060 --> 00:07:40.279 minus t, we measure it against our new fractal 00:07:40.279 --> 00:07:42.410 time. Okay, so the denominator becomes, looking 00:07:42.410 --> 00:07:44.829 at the equation, it becomes t1 to the alpha minus 00:07:44.829 --> 00:07:48.370 t to the alpha as t1 approaches t. So we are 00:07:48.370 --> 00:07:50.329 comparing the change in whatever function we're 00:07:50.329 --> 00:07:52.589 looking at, not to the ticking of a standard 00:07:52.589 --> 00:07:54.829 wall clock, but to the ticking of this newly 00:07:54.829 --> 00:07:57.170 scaled fractal clock. Exactly. And the really 00:07:57.170 --> 00:07:59.170 deep mathematical motivation for this actually 00:07:59.170 --> 00:08:00.949 comes from something called the Taylor series. 00:08:01.209 --> 00:08:05.180 Oh man, the Taylor series. I vaguely remember 00:08:05.180 --> 00:08:07.939 that being the way we approximate really complex 00:08:07.939 --> 00:08:10.800 curves by just layering simple polynomials together. 00:08:11.060 --> 00:08:13.839 That's the one. Let's use an analogy. Think about 00:08:13.839 --> 00:08:16.819 early 3D video game graphics from the 90s. Oh, 00:08:16.879 --> 00:08:19.180 yeah, super low poly count like the original 00:08:19.180 --> 00:08:22.160 Lara Croft with a triangular nose. Right. Standard 00:08:22.160 --> 00:08:24.500 calculus is basically like trying to render a 00:08:24.500 --> 00:08:28.319 highly complex organic face using big flat triangles. 00:08:29.019 --> 00:08:32.690 It's a functional approximation. We say any smooth 00:08:32.690 --> 00:08:34.750 function can be approximated by a sum of straight 00:08:34.750 --> 00:08:37.809 lines. But in the fractal world, straight lines 00:08:37.809 --> 00:08:40.649 simply do not fit. They don't hug the roughness 00:08:40.649 --> 00:08:42.809 of the reality. So the standard triangles are 00:08:42.809 --> 00:08:45.509 just too big. Or, rather, they're entirely the 00:08:45.509 --> 00:08:48.049 wrong shape to begin with. If you try to build 00:08:48.049 --> 00:08:50.429 a perfectly round ball out of square Lego bricks, 00:08:50.830 --> 00:08:52.769 it's always going to look blocky. But if you 00:08:52.769 --> 00:08:55.250 have special curved Lego bricks, or in our case, 00:08:55.350 --> 00:08:57.230 fractal bricks, you get a significantly better 00:08:57.230 --> 00:09:00.049 fit. Right. So researchers generalized the standard 00:09:00.049 --> 00:09:02.360 Taylor series. Instead of expanding with straight 00:09:02.360 --> 00:09:05.320 lines, we approximate the function using terms 00:09:05.320 --> 00:09:08.179 like x to the alpha minus x zero to the alpha. 00:09:08.360 --> 00:09:10.220 So we are essentially upgrading our underlying 00:09:10.220 --> 00:09:13.159 graphics engine. We're moving from PS1 straight 00:09:13.159 --> 00:09:15.980 flat lines to a high definition fractal rendering 00:09:15.980 --> 00:09:18.299 engine. That is a perfect analogy for what the 00:09:18.299 --> 00:09:22.159 math is doing. And this specific upgrade leads 00:09:22.159 --> 00:09:24.159 to what the source is called the fractal Maclaurin 00:09:24.159 --> 00:09:26.440 series. It gives us a way to take a function 00:09:26.440 --> 00:09:28.779 that has fractal support, meaning it literally 00:09:28.779 --> 00:09:31.720 lives on a fractal set and expand it. And then 00:09:31.720 --> 00:09:33.559 let's just treat it with the exact same level 00:09:33.559 --> 00:09:36.539 of rigorous mathematical analysis as we treat 00:09:36.539 --> 00:09:39.539 standard, smooth functions. We aren't just guessing 00:09:39.539 --> 00:09:42.000 or throwing our hands up anymore. We have a highly 00:09:42.000 --> 00:09:44.379 systematic toolkit to break down these rough 00:09:44.379 --> 00:09:47.000 functions. Now, whenever we learn new math rules, 00:09:47.259 --> 00:09:49.340 the absolute first question for anyone doing 00:09:49.340 --> 00:09:52.009 the homework is always, Do the old shortcuts 00:09:52.009 --> 00:09:54.429 still work? Like the chain rule. The chain rule 00:09:54.429 --> 00:09:56.950 absolutely saved my life in calculus. Does that 00:09:56.950 --> 00:09:59.169 survive the transition into the fractal world? 00:09:59.529 --> 00:10:01.809 It does survive, but again with a very specific 00:10:01.809 --> 00:10:04.070 twist. The papers show that if you want to take 00:10:04.070 --> 00:10:07.389 the fractal derivative of a function f with respect 00:10:07.389 --> 00:10:10.269 to x to the alpha, you can absolutely still use 00:10:10.269 --> 00:10:12.429 the standard chain rule, but you have to multiply 00:10:12.429 --> 00:10:14.970 it by a scaling factor. Okay, I see the formula 00:10:14.970 --> 00:10:16.929 right here in the text. It says the fractal derivative 00:10:16.929 --> 00:10:20.009 is equal to 1 over alpha times x to the 1 minus 00:10:20.009 --> 00:10:22.889 alpha, all multiplied by the standard derivative. 00:10:23.230 --> 00:10:25.389 Right, and really think about what that equation 00:10:25.389 --> 00:10:29.210 means physically. It acts as a bridge. A bridge. 00:10:29.370 --> 00:10:32.470 Yeah, it links the strange, jagged fractal derivative 00:10:32.470 --> 00:10:35.070 directly back to the standard smooth derivative 00:10:35.070 --> 00:10:37.909 we already know and love. It mathematically proves 00:10:37.909 --> 00:10:40.429 that these aren't two completely isolated separate 00:10:40.429 --> 00:10:43.090 universes. They are intimately connected by that 00:10:43.090 --> 00:10:46.250 exact scaling factor. I see. It's like that exchange 00:10:46.250 --> 00:10:48.250 rate we mentioned earlier. You can still use 00:10:48.250 --> 00:10:50.970 your US dollars, your standard derivative, but 00:10:50.970 --> 00:10:53.529 you have to pay the exchange fee, which is that 00:10:53.529 --> 00:10:56.210 x to the 1 minus alpha term, to actually spend 00:10:56.210 --> 00:10:59.240 them. in the fractal country. Exactly. And establishing 00:10:59.240 --> 00:11:01.559 that mathematical bridge is incredibly crucial 00:11:01.559 --> 00:11:03.980 if you actually want to apply this theory to 00:11:03.980 --> 00:11:06.779 real life. You can't just stay in abstract pure 00:11:06.779 --> 00:11:09.379 math land forever. Which perfectly brings us 00:11:09.379 --> 00:11:12.460 to the killer app of fractal derivatives. The 00:11:12.460 --> 00:11:15.360 whole reason this isn't just a fun abstract puzzle 00:11:15.360 --> 00:11:18.139 for pure mathematicians to play with. Anomalous 00:11:18.139 --> 00:11:20.679 diffusion. Anomalous diffusion. We've touched 00:11:20.679 --> 00:11:23.279 on this a bit, like pollution seeping into an 00:11:23.279 --> 00:11:26.200 aquifer or heat moving through a weird composite 00:11:26.200 --> 00:11:28.899 material. But let's get really specific for a 00:11:28.899 --> 00:11:32.259 second. Why does the standard model fundamentally 00:11:32.259 --> 00:11:34.740 fail here? Well, the standard model, which is 00:11:34.740 --> 00:11:37.100 usually governed by Fick's second law, assumes 00:11:37.100 --> 00:11:39.299 that things always spread out at a normal, totally 00:11:39.299 --> 00:11:43.559 predictable rate. But in porous media, you encounter 00:11:43.559 --> 00:11:46.899 what statisticians call heavy tails. Heavy tails. 00:11:46.899 --> 00:11:49.000 What does that look like? Imagine that standard 00:11:49.000 --> 00:11:52.360 bell curve again. In a normal distribution, the 00:11:52.360 --> 00:11:54.460 probability of finding a particle really far 00:11:54.460 --> 00:11:56.919 away from the center drops off to zero incredibly 00:11:56.919 --> 00:12:00.460 fast. But in anomalous diffusion, that bell curve 00:12:00.460 --> 00:12:02.980 physically gets stretched out. Oh, okay. The 00:12:02.980 --> 00:12:05.200 tails get heavier. Exactly. The sources heavily 00:12:05.200 --> 00:12:07.519 discuss this linear anomalous transport diffusion 00:12:07.519 --> 00:12:10.019 equation. It looks a lot like the standard diffusion 00:12:10.019 --> 00:12:12.259 equation on the page, but instead of standard 00:12:12.259 --> 00:12:14.399 derivatives, we plug our new fractal derivatives 00:12:14.399 --> 00:12:16.639 right into it. Let me look at equation one from 00:12:16.639 --> 00:12:19.960 the notes here. It says, The change in concentration, 00:12:20.240 --> 00:12:23.919 u, over fractal time, t to the alpha, is equal 00:12:23.919 --> 00:12:26.580 to a diffusion coefficient, d times the change 00:12:26.580 --> 00:12:30.460 in space, x to the beta. Yes. And when you actually 00:12:30.460 --> 00:12:32.639 sit down and solve that differential equation, 00:12:32.659 --> 00:12:34.820 and this is truly where the magic happens, you 00:12:34.820 --> 00:12:37.360 do not get a standard bell curve. You get what 00:12:37.360 --> 00:12:39.919 they call a stretched Gaussian kernel. Trying 00:12:39.919 --> 00:12:41.960 to visualize that. It's like someone grabbed 00:12:41.960 --> 00:12:43.740 the far edges of the bell curve and just pulled 00:12:43.740 --> 00:12:45.860 them violently outwards. Yes, or stepped on the 00:12:45.860 --> 00:12:49.250 top and squashed it down. Statically, the fundamental 00:12:49.250 --> 00:12:51.789 solution involves terms like e to the negative 00:12:51.789 --> 00:12:54.909 x to the power of 2 beta, all divided by 4t to 00:12:54.909 --> 00:12:58.470 the alpha. And that specific shape tells us something 00:12:58.470 --> 00:13:00.870 incredibly vital. It tells us that a pollution 00:13:00.870 --> 00:13:03.289 plume might travel much, much faster than standard 00:13:03.289 --> 00:13:06.429 physics expects. That's super diffusion, because 00:13:06.429 --> 00:13:08.169 maybe it found an underground highway and the 00:13:08.169 --> 00:13:11.450 rock cracks. Or, conversely, it might travel 00:13:11.450 --> 00:13:13.970 much slower, sub -diffusion, because it keeps 00:13:13.970 --> 00:13:16.690 getting stuck in dead ends. And there is a specific 00:13:16.690 --> 00:13:19.409 metric for quantifying the speed, right? I'm 00:13:19.409 --> 00:13:20.870 looking at something called the mean squared 00:13:20.870 --> 00:13:23.309 displacement. Yes. Mean squared displacement 00:13:23.309 --> 00:13:26.210 is the absolute gold standard for measuring any 00:13:26.210 --> 00:13:28.809 kind of diffusion. In normal smooth physics, 00:13:29.289 --> 00:13:31.529 mean squared displacement simply grows linearly 00:13:31.529 --> 00:13:34.610 with time. It's just proportional to two to the 00:13:34.610 --> 00:13:36.590 power of one. Right. You wait twice as long, 00:13:36.669 --> 00:13:39.090 the area covered doubles. Very simple. Exactly. 00:13:39.610 --> 00:13:42.070 But here in the fractal world, the notes highlight 00:13:42.070 --> 00:13:45.440 a very specific asymptote. The mean squared displacement 00:13:45.440 --> 00:13:48.019 is proportional to time raised to a highly complex 00:13:48.019 --> 00:13:51.539 power It's over to the power of 3 alpha minus 00:13:51.539 --> 00:13:55.110 alpha beta all over 2 beta Wow 3 alpha minus 00:13:55.110 --> 00:13:57.789 alpha beta divided by 2 beta. That is incredibly 00:13:57.789 --> 00:14:00.850 specific. It is remarkably specific. And that 00:14:00.850 --> 00:14:04.029 exact power law tells you precisely how anomalous 00:14:04.029 --> 00:14:07.070 the diffusion is. If you physically measure your 00:14:07.070 --> 00:14:09.169 alpha and beta parameters, which basically describe 00:14:09.169 --> 00:14:12.009 the inherent fractal nature of your rock or your 00:14:12.009 --> 00:14:13.990 specific turbulence, you can plug them in and 00:14:13.990 --> 00:14:16.070 predict exactly how that pollution plume will 00:14:16.070 --> 00:14:19.029 spread over 10 years. That is an incredibly powerful 00:14:19.029 --> 00:14:21.409 tool for real world environmental engineering. 00:14:21.629 --> 00:14:24.090 So what this all really boils down to is that 00:14:24.090 --> 00:14:26.590 we can actually predict the chaos. We aren't 00:14:26.590 --> 00:14:28.870 just looking at a cracked rock shrugging our 00:14:28.870 --> 00:14:30.769 shoulders and saying, well, it's messy, so who 00:14:30.769 --> 00:14:32.990 knows where the water goes. Exactly. It literally 00:14:32.990 --> 00:14:36.230 turns chaos into a completely calculable problem. 00:14:37.190 --> 00:14:40.210 But the story of this math actually doesn't end 00:14:40.210 --> 00:14:43.470 there because fractal derivatives are, by their 00:14:43.470 --> 00:14:46.970 very definition, completely local. They only 00:14:46.970 --> 00:14:49.789 look at what's happening right here, right now, 00:14:50.230 --> 00:14:53.009 at one specific point in the fractal space. But 00:14:53.009 --> 00:14:55.669 the real world obviously has memory. What happened 00:14:55.669 --> 00:14:58.110 yesterday affects today. It definitely does. 00:14:58.250 --> 00:15:01.090 And that crucial realization leads us to the 00:15:01.090 --> 00:15:04.419 absolute cutting edge of this field. Fractal 00:15:04.419 --> 00:15:06.200 fractional calculus. Now this is the part of 00:15:06.200 --> 00:15:07.740 the reading that sounded almost like science 00:15:07.740 --> 00:15:10.179 fiction to me. We are now combining fractal derivatives 00:15:10.179 --> 00:15:12.299 with fractional derivatives. And I have to just 00:15:12.299 --> 00:15:14.440 stop you right here because these two words sound 00:15:14.440 --> 00:15:17.399 almost entirely identical. Can you clearly distinguish 00:15:17.399 --> 00:15:20.259 between fractal and fractional for us? Oh, it 00:15:20.259 --> 00:15:22.980 is a hugely common point of confusion. Let's 00:15:22.980 --> 00:15:26.080 break it down very simply. A fractal derivative, 00:15:26.100 --> 00:15:29.379 as we've been discussing, deals entirely with 00:15:29.379 --> 00:15:31.889 scaling the local measure. It deals strictly 00:15:31.889 --> 00:15:34.110 with the geometric jaggedness of the space itself. 00:15:34.570 --> 00:15:37.210 It basically answers the question, how physically 00:15:37.210 --> 00:15:39.370 rough is the road right under my tires right 00:15:39.370 --> 00:15:41.870 now? Okay, so I'm writing that down. Fractal 00:15:41.870 --> 00:15:44.330 equals geometric roughness. Right. A fractional 00:15:44.330 --> 00:15:46.830 derivative is entirely different. It is generally 00:15:46.830 --> 00:15:50.429 a non -local operator. It involves taking an 00:15:50.429 --> 00:15:52.450 integral antler over the entire history of a 00:15:52.450 --> 00:15:55.029 system. It answers the question, where exactly 00:15:55.029 --> 00:15:57.029 have you been before you arrived at this current 00:15:57.029 --> 00:15:59.649 point? It fundamentally captures the memory of 00:15:59.649 --> 00:16:02.179 the system. OK, so fractal is about the jaggedness 00:16:02.179 --> 00:16:04.639 of the space, and fractional is about the memory 00:16:04.639 --> 00:16:07.559 of the past. Beautifully put, yes. And fairly 00:16:07.559 --> 00:16:10.080 recently, brilliant researchers like Professor 00:16:10.080 --> 00:16:13.039 Abdana Tangana from South Africa have been working 00:16:13.039 --> 00:16:15.320 tirelessly on actually merging these two massive 00:16:15.320 --> 00:16:18.039 concepts together. They successfully introduced 00:16:18.039 --> 00:16:19.799 what are called fractal -fractional differential 00:16:19.799 --> 00:16:22.340 operators. This honestly sounds like the Avengers 00:16:22.340 --> 00:16:24.919 of calculus. You are taking the space scalar 00:16:24.919 --> 00:16:26.960 and teaming them up with the history keeper. 00:16:27.259 --> 00:16:29.580 That's a surprisingly accurate way to visualize 00:16:29.580 --> 00:16:32.480 it. But Tangana and other mathematicians realize 00:16:32.480 --> 00:16:35.539 that highly complex real -world systems almost 00:16:35.539 --> 00:16:38.519 always have both fractal geometry and intense 00:16:38.519 --> 00:16:41.820 memory effects happening simultaneously. So the 00:16:41.820 --> 00:16:44.139 rigorously created operators that combine them. 00:16:44.559 --> 00:16:46.899 And the source material lists three very specific 00:16:46.899 --> 00:16:49.259 ways. three distinct kernels to actually compute 00:16:49.259 --> 00:16:51.659 this. Right, I saw these in the text. The power 00:16:51.659 --> 00:16:54.080 law kernel, the exponentially decaying kernel, 00:16:54.500 --> 00:16:57.120 and the generalized Mideg -Leffler kernel. That 00:16:57.120 --> 00:16:59.000 last one sounds like an absolute heavy hitter. 00:16:59.100 --> 00:17:01.460 Oh, it really is. The Mideg -Leffler function 00:17:01.460 --> 00:17:03.919 is essentially considered the queen of fractional 00:17:03.919 --> 00:17:06.400 calculus. But the underlying reason we even have 00:17:06.400 --> 00:17:08.619 three different kernels is what's truly fascinating. 00:17:08.980 --> 00:17:11.220 It's because mathematical memory isn't all exactly 00:17:11.220 --> 00:17:14.259 the same. How so? Like, does math seriously have 00:17:14.259 --> 00:17:16.839 different distinct types of amnesia? In a very 00:17:16.839 --> 00:17:19.720 real way, yes. Think about how human beings forget 00:17:19.720 --> 00:17:22.619 things. I try hard not to. Well, let's look at 00:17:22.619 --> 00:17:25.579 the power law, Colonel. It describes a system 00:17:25.579 --> 00:17:28.759 with what we call long memory. The effects of 00:17:28.759 --> 00:17:32.130 the past do decay over time. But they decay very, 00:17:32.150 --> 00:17:34.650 very slowly. They stick around for ages. This 00:17:34.650 --> 00:17:37.089 is extremely common when you're modeling things 00:17:37.089 --> 00:17:39.630 on vast geological time scales. Okay, so it's 00:17:39.630 --> 00:17:41.670 like a grudge that just never quite goes away. 00:17:41.950 --> 00:17:45.150 Exactly. Now, compare that to the exponentially 00:17:45.150 --> 00:17:47.529 decaying kernel. That one describes a memory 00:17:47.529 --> 00:17:49.910 that feeds away relatively fast. It's essentially 00:17:49.910 --> 00:17:51.890 short -term memory. What happened 10 minutes 00:17:51.890 --> 00:17:53.849 ago matters a lot to the system, but what happened 00:17:53.849 --> 00:17:55.609 yesterday is just effectively gone. And then 00:17:55.609 --> 00:17:57.470 there's the mid -egg -leffler kernel. Right. 00:17:57.609 --> 00:17:59.890 The generalized mid -egg -leffler kernel. That 00:17:59.890 --> 00:18:02.769 describe a highly complex non -local crossover 00:18:02.769 --> 00:18:05.710 between multiple different behaviors. A system 00:18:05.710 --> 00:18:07.690 might actually start with a really fast memory 00:18:07.690 --> 00:18:10.309 decay and then dynamically shift to a slow one 00:18:10.309 --> 00:18:13.049 later on. It is by far the most sophisticated, 00:18:13.049 --> 00:18:16.400 nuanced tool in the entire box. So depending 00:18:16.400 --> 00:18:18.859 entirely on whether you are trying to model a 00:18:18.859 --> 00:18:21.900 slowly drying aquifer or a rapidly spreading 00:18:21.900 --> 00:18:24.759 virus or an unpredictable stock market crash, 00:18:25.259 --> 00:18:28.039 the actual memory of the system functions differently 00:18:28.039 --> 00:18:30.339 and you just pick the specific kernel that fits 00:18:30.339 --> 00:18:32.799 that reality. Exactly. It's absolutely not a 00:18:32.799 --> 00:18:35.339 one -size -fits -all situation. It gives scientists 00:18:35.339 --> 00:18:38.539 a deeply customizable toolkit to describe the 00:18:38.539 --> 00:18:41.799 precise nuance of decay and memory inside a very 00:18:41.799 --> 00:18:44.299 rough fractal environment. I really want to touch 00:18:44.299 --> 00:18:46.599 on this non -local concept just a bit more before 00:18:46.599 --> 00:18:49.279 we wrap up. The deep dive outline explicitly 00:18:49.279 --> 00:18:51.859 mentions fractal non -local calculus, and it 00:18:51.859 --> 00:18:54.039 talks about integrals over a specific range. 00:18:54.220 --> 00:18:56.700 Yes. This directly ties back to the idea that 00:18:56.700 --> 00:18:58.640 in these complex systems, the current state of 00:18:58.640 --> 00:19:00.700 a single point depends heavily on its surrounding 00:19:00.700 --> 00:19:03.180 neighborhood. The strict mathematical definitions 00:19:03.180 --> 00:19:05.660 involve taking integrals over a range, usually 00:19:05.660 --> 00:19:09.099 denoted as a tecus or x to b. And this is honestly 00:19:09.099 --> 00:19:11.730 where the math gets intensely geometric. because 00:19:11.730 --> 00:19:13.730 we are only integrating over what's called the 00:19:13.730 --> 00:19:16.579 perfect fractal set, f to the alpha. Oh, right. 00:19:16.799 --> 00:19:18.819 And the derivative is mathematically defined 00:19:18.819 --> 00:19:20.640 differently depending on whether you are actually 00:19:20.640 --> 00:19:23.440 physically inside that fractal set or in the 00:19:23.440 --> 00:19:25.779 empty gaps between it. Correct. The source text 00:19:25.779 --> 00:19:28.380 explicitly mentions that the derivative is rigorously 00:19:28.380 --> 00:19:31.740 defined via a limit if your point T is actually 00:19:31.740 --> 00:19:35.420 inside the set F. But if E is outside the set, 00:19:35.460 --> 00:19:38.480 meaning it's in the empty void space of the sponge, 00:19:39.019 --> 00:19:41.000 so to speak, the derivative is just mathematically 00:19:41.000 --> 00:19:43.910 zero. That is just wild. It effectively treats 00:19:43.910 --> 00:19:46.190 the empty physical space as if it straight up 00:19:46.190 --> 00:19:48.769 doesn't exist. It absolutely treats the fractal 00:19:48.769 --> 00:19:51.190 structure as the only reality that matters. The 00:19:51.190 --> 00:19:53.890 rough network itself is the entire universe. 00:19:54.069 --> 00:19:56.130 Okay, let's take a deep breath and step back 00:19:56.130 --> 00:19:58.650 for a second. We have covered a massive amount 00:19:58.650 --> 00:20:00.819 of ground today. We essentially started with 00:20:00.819 --> 00:20:04.200 a sponge or porous media and realized that our 00:20:04.200 --> 00:20:07.259 standard trusted speedometers and standard clocks 00:20:07.259 --> 00:20:10.400 are dx and dt. They just flat out do not work 00:20:10.400 --> 00:20:12.680 in there. They give us completely false, useless 00:20:12.680 --> 00:20:15.140 readings. So mathematics had to invent a totally 00:20:15.140 --> 00:20:18.279 new clock, t to the alpha, and a totally new 00:20:18.279 --> 00:20:21.579 ruler, x to the beta. We completely rebuilt the 00:20:21.579 --> 00:20:23.500 Taylor series from the ground up to fit this 00:20:23.500 --> 00:20:25.599 new world, upgrading our graphics engine, as 00:20:25.599 --> 00:20:28.299 we joked earlier. And we found out that doing 00:20:28.299 --> 00:20:31.160 all of this allows us to finally model how things 00:20:31.160 --> 00:20:34.460 like pollution practically spread in an aquifer, 00:20:34.720 --> 00:20:38.099 using these weird stretched Gaussian curves instead 00:20:38.099 --> 00:20:40.579 of standard bell curves. Which is absolutely 00:20:40.579 --> 00:20:43.220 vital for public safety and actual civil engineering. 00:20:43.400 --> 00:20:45.859 And then, just to push it over the edge, we realized 00:20:45.859 --> 00:20:48.299 that we can physically combine the space scaling 00:20:48.299 --> 00:20:50.869 math with fractional calculus to completely account 00:20:50.869 --> 00:20:54.450 for a complex system's memory using intense tools 00:20:54.450 --> 00:20:57.009 like the Mitakag -Luffler kernel to decide if 00:20:57.009 --> 00:20:59.509 a system has a long -term grudge or just short 00:20:59.509 --> 00:21:02.170 -term memory. That is an absolutely excellent 00:21:02.170 --> 00:21:04.529 summary of the entire framework. We've literally 00:21:04.529 --> 00:21:06.549 gone from making a basic observation that rocks 00:21:06.549 --> 00:21:09.250 are rough to developing a highly sophisticated, 00:21:09.589 --> 00:21:12.329 robust mathematical language fractal fractional 00:21:12.329 --> 00:21:15.170 calculus that can accurately describe the complex 00:21:15.170 --> 00:21:17.410 flow of energy and matter straight through that 00:21:17.410 --> 00:21:19.799 roughness. It's just so impressive to me because 00:21:19.799 --> 00:21:23.079 it proves that math isn't just a static dead 00:21:23.079 --> 00:21:25.559 thing. We always kind of think of calculus as 00:21:25.559 --> 00:21:28.420 this immutable stone tablet that was handed down 00:21:28.420 --> 00:21:31.359 by Newton and Leibniz centuries ago, but it's 00:21:31.359 --> 00:21:33.240 actually actively evolving right now to meet 00:21:33.240 --> 00:21:35.859 the sheer complexity of the real world we keep 00:21:35.859 --> 00:21:39.140 discovering. That is honestly the absolute most 00:21:39.140 --> 00:21:41.680 important takeaway for me from all these sources. 00:21:42.119 --> 00:21:44.460 The fractal derivative is not just some abstract 00:21:44.460 --> 00:21:47.299 mental gymnastics for academics. It is a strict 00:21:47.279 --> 00:21:50.599 necessary evolution. If we ever want to truly 00:21:50.599 --> 00:21:52.839 understand the real world, which, let's face 00:21:52.839 --> 00:21:56.140 it, is rarely ever smooth and simple, we absolutely 00:21:56.140 --> 00:21:58.140 need mathematics that speaks the native language 00:21:58.140 --> 00:22:01.180 of roughness. We have to be able to model chaos 00:22:01.180 --> 00:22:04.339 and turbulence and anomalous diffusion accurately, 00:22:04.640 --> 00:22:06.740 or else our bridges fall down and our climate 00:22:06.740 --> 00:22:08.680 models completely fail. It effectively gives 00:22:08.680 --> 00:22:11.160 us the lens to finally see the underlying order 00:22:11.160 --> 00:22:13.720 hidden deep inside the rough edges. It does. 00:22:13.799 --> 00:22:15.500 It really does. And, you know, if I can just 00:22:15.500 --> 00:22:17.140 leave you with one final thought that pushes 00:22:17.140 --> 00:22:19.609 this entire concept just a little bit. Oh, please 00:22:19.609 --> 00:22:21.730 do. This is where it always gets really interesting. 00:22:22.509 --> 00:22:25.130 So the sources mention a specific paper by Wen 00:22:25.130 --> 00:22:28.930 Chen, and the title is, Time -Space Fabric Underlying 00:22:28.930 --> 00:22:32.950 Anomalous Diffusion. It actively suggests that 00:22:32.950 --> 00:22:35.849 the mere existence of anomalous diffusion implies 00:22:35.849 --> 00:22:38.490 that there is an underlying fractal time -space 00:22:38.490 --> 00:22:41.910 fabric to reality. A fractal time -space fabric. 00:22:42.210 --> 00:22:43.529 Really think about the implications of that. 00:22:43.869 --> 00:22:47.150 We universally assume that time just flows exactly 00:22:47.150 --> 00:22:49.450 the same for everyone everywhere at a perfectly 00:22:49.450 --> 00:22:51.890 constant, smooth rate. Right, time is time. But 00:22:51.890 --> 00:22:54.009 in these incredibly complex, porous systems, 00:22:54.509 --> 00:22:56.529 the physical math works best if we fundamentally 00:22:56.529 --> 00:22:59.309 assume that time itself actually scales differently 00:22:59.309 --> 00:23:01.730 depending on the specific fractal dimension of 00:23:01.730 --> 00:23:04.509 the space. Whoa! If time literally scales as 00:23:04.509 --> 00:23:06.650 t to the alpha when you're inside a turbulent 00:23:06.650 --> 00:23:09.950 cloud, What does that imply about our basic human 00:23:09.950 --> 00:23:12.470 perception of flow and causality and change when 00:23:12.470 --> 00:23:15.309 we look at complex systems? So in a very real 00:23:15.309 --> 00:23:17.869 mathematical way, the actual clock of the universe 00:23:17.869 --> 00:23:19.890 might tick differently depending entirely on 00:23:19.890 --> 00:23:22.269 just how inherently rough the neighborhood is. 00:23:22.309 --> 00:23:24.809 It's a deeply provocative thought. The math definitely 00:23:24.809 --> 00:23:27.529 points in that direction. Well, that is definitely 00:23:27.529 --> 00:23:29.630 something for all of you to just stare at the 00:23:29.630 --> 00:23:32.269 ceiling and think about tonight. Thank you so 00:23:32.269 --> 00:23:35.289 much for guiding us through the incredibly jagged 00:23:35.289 --> 00:23:38.089 complex world of the fractal derivative today. 00:23:38.309 --> 00:23:40.250 It was my absolute pleasure. It's always a really 00:23:40.250 --> 00:23:43.269 fun time exploring the rough edges of reality. 00:23:43.529 --> 00:23:45.869 And a huge thank you to everyone listening for 00:23:45.869 --> 00:23:48.069 taking the plunge into the math of the rough 00:23:48.069 --> 00:23:50.089 and jagged with us. We will catch you on the 00:23:50.089 --> 00:23:50.869 next deep dive.