WEBVTT 00:00:00.000 --> 00:00:03.120 Welcome back to the Deep Dive. Today we are standing 00:00:03.120 --> 00:00:05.719 on the edge. Ideally, not literally. No, not 00:00:05.719 --> 00:00:07.280 literally, but metaphorically. We're talking 00:00:07.280 --> 00:00:09.720 about boundaries. You know that moment where 00:00:09.720 --> 00:00:13.279 a system just shifts? Right. Like on one side, 00:00:13.380 --> 00:00:15.400 everything is vibrating and energetic, moving 00:00:15.400 --> 00:00:17.519 like a wave, and then you cross this invisible 00:00:17.519 --> 00:00:20.640 line and suddenly everything just changes. It 00:00:20.640 --> 00:00:24.940 gets quiet. It decays. That is a very poetic 00:00:24.940 --> 00:00:26.780 way to describe what's called a turning point. 00:00:27.149 --> 00:00:29.769 I try. But seriously, we are looking at a specific 00:00:29.769 --> 00:00:32.429 mathematical concept that describes exactly that 00:00:32.429 --> 00:00:35.390 transition. It's called the airy function. And 00:00:35.390 --> 00:00:38.929 I know you just heard function and might be reaching 00:00:38.929 --> 00:00:41.189 for the skip button, but don't do it. Because 00:00:41.189 --> 00:00:43.250 this isn't just dry calculus. This is the math 00:00:43.250 --> 00:00:45.689 of rainbows. It is. And it's the math of quantum 00:00:45.689 --> 00:00:49.270 mechanics and these strange beams of light that 00:00:49.270 --> 00:00:52.149 seem to curve all on their own. It is a surprisingly 00:00:52.149 --> 00:00:54.530 beautiful bridge between two very different worlds 00:00:54.530 --> 00:00:57.130 of physics. So we have a stack of notes here, 00:00:57.270 --> 00:00:59.689 primarily based on the Wikipedia entry for the 00:00:59.689 --> 00:01:01.509 Airy function, which actually goes into some 00:01:01.509 --> 00:01:03.590 serious depth on the history and the heavy math. 00:01:04.129 --> 00:01:07.650 Our mission today is to decode this. We want 00:01:07.650 --> 00:01:10.790 to understand how one little equation acts as 00:01:10.790 --> 00:01:12.510 this bridge you mentioned. So let's start with 00:01:12.510 --> 00:01:15.430 the name, Airy. Is it named that because it describes 00:01:15.430 --> 00:01:19.129 air? It's actually named after a person, Sir 00:01:19.129 --> 00:01:22.709 George Bedell Airy. Oh, okay. He was the British 00:01:22.709 --> 00:01:25.620 Astronomer Royal back in the 19th century. Okay. 00:01:25.900 --> 00:01:28.120 Astronomer Royal. That's a cool title. It is. 00:01:28.439 --> 00:01:31.379 And like a lot of scientists of his time, he 00:01:31.379 --> 00:01:33.280 wasn't just looking at stars. He was obsessed 00:01:33.280 --> 00:01:36.560 with optics, how light behaves. Before we get 00:01:36.560 --> 00:01:38.500 to the history, we really have to look at the 00:01:38.500 --> 00:01:41.099 beast itself, the equation. Right. The Airy equation 00:01:41.099 --> 00:01:43.459 or the Stokes equation, as it's sometimes called. 00:01:43.519 --> 00:01:45.579 I'm looking at it here and it looks deceptively 00:01:45.579 --> 00:01:49.180 simple. It's just y double prime minus xy equals 00:01:49.180 --> 00:01:53.000 zero. Exactly. yxy equals zero. Exactly. yxy 00:01:53.000 --> 00:01:56.430 equals zero. Okay, let's unpack this. For anyone 00:01:56.430 --> 00:01:58.489 who hasn't done calculus in a while, y double 00:01:58.489 --> 00:02:00.609 prime, that's the second derivative, that describes 00:02:00.609 --> 00:02:03.950 curvature, right? Or acceleration. Correct. In 00:02:03.950 --> 00:02:06.010 physics, the second derivative tells you how 00:02:06.010 --> 00:02:08.490 the function is bending. If you think of a graph, 00:02:08.629 --> 00:02:11.830 it's the concavity. So the equation says that 00:02:11.830 --> 00:02:14.330 this curvature is equal to x times the value 00:02:14.330 --> 00:02:16.750 of the function itself. And that simple relationship 00:02:16.750 --> 00:02:19.830 creates this whole turning point behavior. It 00:02:19.830 --> 00:02:21.689 does. And this is why mathematicians love it. 00:02:21.879 --> 00:02:24.719 It is considered the simplest second -order linear 00:02:24.719 --> 00:02:27.080 differential equation that has a turning point. 00:02:27.599 --> 00:02:30.860 But to see why, you have to sort of play with 00:02:30.860 --> 00:02:33.560 the variable x. Okay, I'm game. Let's say x is 00:02:33.560 --> 00:02:34.919 a negative number. We're on the left side of 00:02:34.919 --> 00:02:37.759 the graph. If x is negative, say, minus 5, then 00:02:37.759 --> 00:02:40.639 the equation becomes y. double prime equals negative 00:02:40.639 --> 00:02:45.180 five y or y double prime plus five y equals zero 00:02:45.180 --> 00:02:47.120 wait i recognize that structure acceleration 00:02:47.120 --> 00:02:49.639 equals negative position that's that's a spring 00:02:49.639 --> 00:02:52.219 exactly it's the equation for a simple harmonic 00:02:52.219 --> 00:02:54.879 oscillator a spring bouncing back and forth, 00:02:54.919 --> 00:02:57.539 or a wave, the curvature is always pulling the 00:02:57.539 --> 00:02:59.639 function back toward zero. So when you look at 00:02:59.639 --> 00:03:01.659 the graph of the airy function on the left side, 00:03:01.699 --> 00:03:04.379 where x is negative, it's wiggly. It oscillates. 00:03:04.840 --> 00:03:07.180 Precisely. But notice what happens as you go 00:03:07.180 --> 00:03:09.939 further left into more negative numbers. The 00:03:09.939 --> 00:03:13.599 spring gets stiffer, you could say, because x 00:03:13.599 --> 00:03:15.680 is a bigger negative number. So the waves get 00:03:15.680 --> 00:03:18.500 faster. Yes. The frequency increases, and the 00:03:18.500 --> 00:03:20.840 amplitude, the height of the waves, gets smaller. 00:03:21.000 --> 00:03:24.520 It's a nervous, fast ripple. Okay, so on the 00:03:24.520 --> 00:03:27.159 left, we have ripples. Now, let's cross that 00:03:27.159 --> 00:03:30.419 line. What happens when we hit zero, when x becomes 00:03:30.419 --> 00:03:33.639 positive? That is the turning point. At x equals 00:03:33.639 --> 00:03:37.139 zero, the sign flips. Now you have y double prime 00:03:37.139 --> 00:03:40.180 equals xy, where x is positive. So the curvature 00:03:40.180 --> 00:03:42.719 isn't pulling it back to zero anymore. It's pushing 00:03:42.719 --> 00:03:44.939 it away. Exactly. If the function is positive, 00:03:45.120 --> 00:03:47.099 the curvature bends it more positive. If it's 00:03:47.099 --> 00:03:48.800 negative, it bends it more negative. It stops 00:03:48.800 --> 00:03:50.599 oscillating. It becomes an exponential function. 00:03:50.960 --> 00:03:53.219 And exponential usually means it just takes off, 00:03:53.219 --> 00:03:55.819 right, or crashes. In the case of the main area 00:03:55.819 --> 00:03:58.360 function, which we call AI of X, it crashes. 00:03:58.419 --> 00:04:01.039 It decays rapidly. It just slides down towards 00:04:01.039 --> 00:04:03.360 zero and fades away. So if you were to draw this, 00:04:03.460 --> 00:04:05.300 on the left you have these ripples, and then 00:04:05.300 --> 00:04:08.520 they hit this invisible wall at zero, and the 00:04:08.520 --> 00:04:11.080 ripples just smooth out and die off into a long 00:04:11.080 --> 00:04:14.120 tail on the right. That is the perfect visualization. 00:04:14.919 --> 00:04:17.079 And that is why it's so important in physics. 00:04:17.399 --> 00:04:20.550 Because in the real world... Boundaries are rarely 00:04:20.550 --> 00:04:24.230 hard. If a wave hits a wall, it doesn't just 00:04:24.230 --> 00:04:27.129 stop instantly at the atomic level. It penetrates 00:04:27.129 --> 00:04:29.810 a little bit. It decays. The Airy function describes 00:04:29.810 --> 00:04:33.490 that soft boundary where oscillation turns into 00:04:33.490 --> 00:04:36.250 decay. That is fascinating. It's like the mathematical 00:04:36.250 --> 00:04:39.910 description of fizzling out. In a way, yes. But 00:04:39.910 --> 00:04:42.129 here is where it gets really interesting. Differential 00:04:42.129 --> 00:04:44.889 equations like this usually have two solutions. 00:04:45.089 --> 00:04:47.149 We call them linearly independent solutions. 00:04:47.550 --> 00:04:49.389 Right. I saw this in the notes. We have AI of 00:04:49.389 --> 00:04:51.050 X, which is the good one we just talked about. 00:04:51.089 --> 00:04:53.189 But then there's an evil twin. I wouldn't call 00:04:53.189 --> 00:04:55.910 it evil, but it is certainly unruly. The first 00:04:55.910 --> 00:04:58.629 solution is AI of X. It's well behaved because 00:04:58.629 --> 00:05:01.050 as X gets bigger, it goes to zero. That makes 00:05:01.050 --> 00:05:03.269 it physically useful. For things like probability 00:05:03.269 --> 00:05:06.209 or particles trapped in a box. Exactly. And the 00:05:06.209 --> 00:05:08.350 second one? The second one is called B of X. 00:05:08.889 --> 00:05:12.709 B -I. B -I. Okay, what does B do? On the negative 00:05:12.709 --> 00:05:16.110 side, the wave side, B looks just like A -I. 00:05:16.189 --> 00:05:18.790 It oscillates with the same amplitude, just shifted 00:05:18.790 --> 00:05:22.000 in phase a bit. But on the positive side... where 00:05:22.000 --> 00:05:26.160 AI politely fades to zero, B explodes. Explodes. 00:05:26.160 --> 00:05:28.579 It grows exponentially to infinity. Oh, wow. 00:05:28.740 --> 00:05:31.360 So if AI is the fizzle, B is the kaboom. Basically. 00:05:31.579 --> 00:05:33.879 And because it shoots off to infinity, it's often 00:05:33.879 --> 00:05:36.699 discarded in simple physical problems where you 00:05:36.699 --> 00:05:39.300 need a finite answer. You can't have a probability 00:05:39.300 --> 00:05:41.860 of finding a particle that is infinity, you know? 00:05:41.920 --> 00:05:43.439 So why do we keep it around? Why not just delete 00:05:43.439 --> 00:05:45.259 it from the textbooks? Because mathematically, 00:05:45.500 --> 00:05:47.720 you need it to form a basis. You can't define 00:05:47.720 --> 00:05:49.819 the general solution to the differential equation 00:05:49.819 --> 00:05:53.519 without it. And sometimes you are describing 00:05:53.519 --> 00:05:56.779 a system that grows. So B is the necessary partner, 00:05:56.860 --> 00:05:59.740 even if he's a bit loud. Got it. The unruly sibling 00:05:59.740 --> 00:06:02.660 stays in the picture. Now let's pivot back to 00:06:02.660 --> 00:06:05.360 the man himself, George Bedell Airy. You said 00:06:05.360 --> 00:06:07.240 he didn't invent this just to torture calculus 00:06:07.240 --> 00:06:10.259 students. He was looking at rainbows. He was. 00:06:10.379 --> 00:06:13.019 And not just the cartoon Rainbow. He was studying 00:06:13.019 --> 00:06:16.819 something called the supernumerary rainbow. Supernumerary. 00:06:17.180 --> 00:06:20.040 That sounds expensive. It does. Have you ever 00:06:20.040 --> 00:06:22.420 looked at a really bright rainbow and noticed 00:06:22.420 --> 00:06:25.199 that just inside the main violet band there are 00:06:25.199 --> 00:06:28.600 these faint repeating pink and green bands? I 00:06:28.600 --> 00:06:30.860 think so, like echoes of the rainbow. Exactly, 00:06:31.060 --> 00:06:34.019 those are supernumerary rainbows. Classical geometric 00:06:34.019 --> 00:06:37.300 optics just drawing lines for light rays couldn't 00:06:37.300 --> 00:06:39.779 explain them. According to simple ray theory, 00:06:39.920 --> 00:06:41.480 the light should just stop at the edge of the 00:06:41.480 --> 00:06:44.600 rainbow. But it doesn't. No, light is a wave 00:06:44.600 --> 00:06:47.980 and waves interfere with each other. Ari was 00:06:47.980 --> 00:06:50.079 trying to calculate the intensity of light near 00:06:50.079 --> 00:06:53.019 a caustic. Define caustic for us, because I usually 00:06:53.019 --> 00:06:56.740 think of caustic wit or acid. In optics, a caustic 00:06:56.740 --> 00:06:58.720 is an envelope of light rays. It's where light 00:06:58.720 --> 00:07:01.160 rays bunch up and focus. You know when you shine 00:07:01.160 --> 00:07:02.720 a light through a glass of water and you see 00:07:02.720 --> 00:07:04.980 that bright curved shape on the table? Yeah, 00:07:05.019 --> 00:07:07.439 that sharp curve of light. That's a caustic. 00:07:07.439 --> 00:07:10.220 A rainbow is essentially a giant caustic in the 00:07:10.220 --> 00:07:14.110 sky. It's a fold in the light intensity. Airy 00:07:14.110 --> 00:07:16.410 realized that near this edge, the intensity of 00:07:16.410 --> 00:07:19.410 the light doesn't just cut off. It oscillates 00:07:19.410 --> 00:07:22.250 and then fades. Wait, oscillates and then fades? 00:07:22.410 --> 00:07:24.769 That sounds familiar. It is exactly the Airy 00:07:24.769 --> 00:07:26.990 function. He developed this math specifically 00:07:26.990 --> 00:07:29.230 to describe how the brightness of the rainbow 00:07:29.230 --> 00:07:32.209 ripples near that edge. That is so cool. So he 00:07:32.209 --> 00:07:35.720 writes this down in... When was this? 1838. 1838. 00:07:35.720 --> 00:07:37.579 He writes down the math. Was he right? He was. 00:07:37.699 --> 00:07:41.079 A few years later, in 1841, a physicist named 00:07:41.079 --> 00:07:43.920 William Hallows Miller tested it. He didn't have 00:07:43.920 --> 00:07:46.300 a convenient rainstorm, so he used a thin cylinder 00:07:46.300 --> 00:07:48.720 of water to simulate a raindrop and shone light 00:07:48.720 --> 00:07:51.860 through it. He observed 30 of these tiny dark 00:07:51.860 --> 00:07:54.199 and bright bands, these supernumerary bands, 00:07:54.500 --> 00:07:57.339 and they matched Airy's theory perfectly. I love 00:07:57.339 --> 00:08:00.060 that. 30 bands. That's precision. It is. And 00:08:00.060 --> 00:08:02.620 interestingly, Airy didn't call it the Airy function 00:08:02.620 --> 00:08:05.600 at the time. The notation AI of X was actually 00:08:05.600 --> 00:08:07.720 introduced much later by a geophysicist named 00:08:07.720 --> 00:08:10.740 Harold Jeffries, but the name stuck to Airy. 00:08:10.779 --> 00:08:14.680 As it should. Okay, so we have rainbows. That's 00:08:14.680 --> 00:08:17.779 a... Beautiful classical application. But the 00:08:17.779 --> 00:08:19.839 outline says this function is a heavy hitter 00:08:19.839 --> 00:08:21.839 in quantum mechanics, too. Oh, absolutely. In 00:08:21.839 --> 00:08:23.939 fact, for a modern physics student, the Airy 00:08:23.939 --> 00:08:26.220 function is almost synonymous with quantum mechanics. 00:08:26.439 --> 00:08:28.540 Why? What's the connection? Think back to the 00:08:28.540 --> 00:08:31.220 shape of the function. Ripples on one side, decay 00:08:31.220 --> 00:08:33.860 on the other. In quantum mechanics, particles 00:08:33.860 --> 00:08:37.480 behave like waves. Right. Wave -particle duality. 00:08:37.580 --> 00:08:40.500 So imagine a particle trapped in a potential 00:08:40.500 --> 00:08:44.710 well. Specifically, a triangular well. Imagine 00:08:44.710 --> 00:08:48.049 a floor that is sloped, like a ramp, and a hard 00:08:48.049 --> 00:08:50.669 wall on one side. Okay, so the particle can bounce 00:08:50.669 --> 00:08:52.669 against the wall, but as it goes up the ramp, 00:08:52.830 --> 00:08:54.990 a force slows it down until it stops and slides 00:08:54.990 --> 00:08:58.549 back. Exactly. Classically, there is a specific 00:08:58.549 --> 00:09:00.629 point on the ramp where the particle stops and 00:09:00.629 --> 00:09:03.129 turns around. A turning point. There's that phrase 00:09:03.129 --> 00:09:05.509 again. In quantum mechanics, the particle is 00:09:05.509 --> 00:09:08.059 a wave. On the ramp side, before the turning 00:09:08.059 --> 00:09:10.759 point, the particle has energy, so its wave function 00:09:10.759 --> 00:09:13.399 oscillates. That's the Wigley part of AO of X. 00:09:13.539 --> 00:09:15.919 But past the turning point, where classically 00:09:15.919 --> 00:09:18.620 it shouldn't be able to go. It tunnels. It tunnels. 00:09:18.740 --> 00:09:21.259 It penetrates into the forbidden region. And 00:09:21.259 --> 00:09:24.139 that penetration is perfectly described by the 00:09:24.139 --> 00:09:26.779 decay tail of the Airy function. So the Airy 00:09:26.779 --> 00:09:29.240 function is literally the map of a particle hitting 00:09:29.240 --> 00:09:32.360 a soft wall. Exactly. It solves the Schrodinger 00:09:32.360 --> 00:09:35.100 equation for a particle in a constant force field 00:09:35.100 --> 00:09:38.159 like a constant electric field. It is the textbook 00:09:38.159 --> 00:09:40.379 description of a boundary in quantum mechanics. 00:09:40.659 --> 00:09:41.980 This brings me to something else in the notes. 00:09:42.100 --> 00:09:46.250 The WKB approximation. The notes call it a uniform 00:09:46.250 --> 00:09:50.370 semi -classical approximation. That is a mouthful. 00:09:50.389 --> 00:09:52.309 What does it actually mean? This is where the 00:09:52.309 --> 00:09:55.210 Airy function does some heavy lifting. In physics, 00:09:55.309 --> 00:09:57.529 we often can't solve equations perfectly. We 00:09:57.529 --> 00:10:00.610 use approximations. The WKB method is a way to 00:10:00.610 --> 00:10:02.929 approximate the wave function when the properties 00:10:02.929 --> 00:10:05.110 of the environment are changing slowly. Okay, 00:10:05.149 --> 00:10:07.350 so it's a shortcut. Yes, and the shortcut works 00:10:07.350 --> 00:10:10.169 great when the particle is moving fast, far from 00:10:10.169 --> 00:10:11.950 the turning point, or when it's deep in the forbidden 00:10:11.950 --> 00:10:15.899 region. But right at the turning point, the approximation 00:10:15.899 --> 00:10:18.700 breaks. Why does it break? Because the approximation 00:10:18.700 --> 00:10:21.559 relies on the momentum of the particle. At the 00:10:21.559 --> 00:10:24.039 turning point, the velocity goes to zero. The 00:10:24.039 --> 00:10:26.259 wavelength effectively tries to become infinite. 00:10:26.440 --> 00:10:29.480 The math just, it blows up. You get a divide 00:10:29.480 --> 00:10:32.019 by zero error. So the shortcut fails exactly 00:10:32.019 --> 00:10:34.240 where the action is happening. Exactly. So physicists 00:10:34.240 --> 00:10:38.019 use the Airy function as a patch. They know that 00:10:38.019 --> 00:10:40.679 locally... Right near the turning point, the 00:10:40.679 --> 00:10:42.919 system looks like the area equation. So they 00:10:42.919 --> 00:10:46.179 solve that little section using AI of X and then 00:10:46.179 --> 00:10:49.059 glue it to the WKB solutions on either side. 00:10:49.220 --> 00:10:51.340 It's the duct tape of quantum mechanics. A very 00:10:51.340 --> 00:10:54.360 elegant, high -precision duct tape, yes. It connects 00:10:54.360 --> 00:10:56.919 the oscillating world to the decaying world smoothly 00:10:56.919 --> 00:10:59.620 without the math blowing up. I love the idea 00:10:59.620 --> 00:11:02.080 of a patching function. Now, I want to talk about 00:11:02.080 --> 00:11:04.000 one application that jumped out at me from the 00:11:04.000 --> 00:11:08.340 notes. This accelerating beam in optics. This 00:11:08.340 --> 00:11:10.279 sounds like sci -fi. This is one of the coolest 00:11:10.279 --> 00:11:12.899 modern applications. It was theoretically proposed 00:11:12.899 --> 00:11:15.899 and then realized in the lab around 2007. Imagine 00:11:15.899 --> 00:11:18.159 a laser beam. Usually light travels in a straight 00:11:18.159 --> 00:11:20.399 line, right? Rule number one of light. Light 00:11:20.399 --> 00:11:23.009 travels straight. Well, scientists created what 00:11:23.009 --> 00:11:26.330 they call a transversely asymmetric optical beam. 00:11:26.649 --> 00:11:28.970 Basically, they shaped the cross section of the 00:11:28.970 --> 00:11:31.629 beam so that the intensity profile looks exactly 00:11:31.629 --> 00:11:34.250 like the airy function. So a bright peak on one 00:11:34.250 --> 00:11:36.669 side and then those fading ripples on the other. 00:11:36.809 --> 00:11:39.629 Exactly. When you launch this beam, something 00:11:39.629 --> 00:11:42.750 bizarre happens. The main brightest peak appears 00:11:42.750 --> 00:11:46.129 to curve. It accelerates to the side. It curves 00:11:46.129 --> 00:11:49.429 like a banana cake in soccer. Yes. It creates 00:11:49.429 --> 00:11:52.090 a parabolic arc. as it travels through free space. 00:11:52.450 --> 00:11:55.970 But physics, conservation of momentum, you can't 00:11:55.970 --> 00:11:58.090 just have light turn left for no reason. It needs 00:11:58.090 --> 00:12:00.190 to bounce off something. Physics is safe, don't 00:12:00.190 --> 00:12:02.549 worry. This is where the tail of the airy function 00:12:02.549 --> 00:12:05.110 comes in. Remember that long tail of ripples 00:12:05.110 --> 00:12:07.210 on the other side? The part that fades out. Yes. 00:12:07.269 --> 00:12:10.210 As the main break peak curves one way, that low 00:12:10.210 --> 00:12:12.690 intensity tail spreads out and curves in the 00:12:12.690 --> 00:12:14.789 opposite direction. Oh, I see. So the average 00:12:14.789 --> 00:12:16.850 position of the light is still going straight. 00:12:17.149 --> 00:12:20.340 Exactly. The center of gravity of the beam travels 00:12:20.340 --> 00:12:23.139 in a straight line, satisfying conservation of 00:12:23.139 --> 00:12:26.379 momentum. But visually, and in terms of where 00:12:26.379 --> 00:12:29.320 the energy is focused, the beam effectively bends 00:12:29.320 --> 00:12:32.519 around corners. That is wild. So you could shoot 00:12:32.519 --> 00:12:34.899 a laser beam around an obstacle? In principle, 00:12:35.000 --> 00:12:37.879 yes. They are called airy beams. They are also 00:12:37.879 --> 00:12:40.279 non -diffracting, meaning they don't spread out 00:12:40.279 --> 00:12:42.379 and get fuzzy as fast as normal laser beams. 00:12:42.519 --> 00:12:45.379 Just because they used this specific math shape. 00:12:45.500 --> 00:12:47.879 Because the airy function has these special properties 00:12:47.879 --> 00:12:50.879 of stability. And get this, it's self -healing. 00:12:51.000 --> 00:12:53.769 Self -healing. Like Wolverine. A little bit. 00:12:53.789 --> 00:12:56.570 If you place a small obstacle in the path and 00:12:56.570 --> 00:12:59.789 block the main bright peak, the tail, because 00:12:59.789 --> 00:13:01.570 it contains the phase information of the whole 00:13:01.570 --> 00:13:03.950 function, will actually reconstruct the main 00:13:03.950 --> 00:13:06.529 peak further down the line. No way. The tail 00:13:06.529 --> 00:13:09.149 grows new head. It does. The interference pattern 00:13:09.149 --> 00:13:11.909 reforms the airy shape. Okay, my mind is officially 00:13:11.909 --> 00:13:14.809 blown. We went from a 19th century astronomer 00:13:14.809 --> 00:13:18.289 looking at rainbows to self -healing curving 00:13:18.289 --> 00:13:20.429 laser beams that fix themselves if you block 00:13:20.429 --> 00:13:22.250 them. And we aren't even done. It shows up in 00:13:22.250 --> 00:13:24.870 probability theory too. Right, the random matrices. 00:13:24.950 --> 00:13:27.409 This seems like a totally different field. We're 00:13:27.409 --> 00:13:29.370 leaving physics now and entering statistics. 00:13:29.570 --> 00:13:32.730 We are. Random matrix theory deals with the statistics 00:13:32.730 --> 00:13:35.789 of huge grids filled with randomness. numbers. 00:13:35.929 --> 00:13:38.450 It turns out, if you look at the distribution 00:13:38.450 --> 00:13:41.289 of the largest eigenvalues, the most important 00:13:41.289 --> 00:13:44.149 numbers describing these matrices, they follow 00:13:44.149 --> 00:13:47.129 a specific pattern. Let me guess. It's airy. 00:13:47.230 --> 00:13:50.049 It's the Tracy -Whidham distribution, which is 00:13:50.049 --> 00:13:54.179 defined using the airy function. Why airy? Where 00:13:54.179 --> 00:13:56.740 is the turning point in a random grid of numbers? 00:13:57.000 --> 00:13:58.980 It represents the soft edge of the spectrum. 00:13:59.220 --> 00:14:01.440 The eigenvalues are packed together like a fluid, 00:14:01.600 --> 00:14:04.259 and at the edge of that fluid, the density drops 00:14:04.259 --> 00:14:06.799 off. That drop -off is described by the Airy 00:14:06.799 --> 00:14:08.700 function. So it's a boundary again. It's always 00:14:08.700 --> 00:14:10.919 a boundary. This pops up in everything from nuclear 00:14:10.919 --> 00:14:13.419 physics to the behavior of the stock market to 00:14:13.419 --> 00:14:16.200 how distinct interface growth happens, like how 00:14:16.200 --> 00:14:18.639 a coffee stain spreads or how a Tetris game fills 00:14:18.639 --> 00:14:21.559 up. The Airy function describes Tetris. In a 00:14:21.559 --> 00:14:24.899 very abstract statistical way via the KPZ equation, 00:14:25.320 --> 00:14:28.580 yes, it describes the fluctuations of the surface 00:14:28.580 --> 00:14:32.440 growth. If you drop blocks randomly, the roughness 00:14:32.440 --> 00:14:35.039 of the top layer follows statistics governed 00:14:35.039 --> 00:14:37.620 by the Airy function. It really is the boundary 00:14:37.620 --> 00:14:40.500 function. Wherever there is an edge, Airy is 00:14:40.500 --> 00:14:43.100 there. That is the big takeaway. Before we wrap 00:14:43.100 --> 00:14:45.820 up, I do want to dip our toes into the deep part 00:14:45.820 --> 00:14:48.399 of the notes, the mathematical nuances. We talked 00:14:48.399 --> 00:14:51.399 about AI and buy, but the notes mentioned Bessel 00:14:51.399 --> 00:14:53.779 functions. Are they related? They are cousins. 00:14:54.080 --> 00:14:56.399 The Airy functions can actually be written in 00:14:56.399 --> 00:14:58.379 terms of Bessel functions of fractional order, 00:14:58.539 --> 00:15:01.019 specifically order one -third. Why one -third? 00:15:01.059 --> 00:15:03.580 Is that arbitrary? No, it comes from the cubic 00:15:03.580 --> 00:15:05.960 nature of the phase. Remember the integral definition 00:15:05.960 --> 00:15:08.220 of the Airy function. It involves t cubed over 00:15:08.220 --> 00:15:11.139 three. That cubic term is the signature of the 00:15:11.139 --> 00:15:13.580 Airy function. When you do the math, that cubic 00:15:13.580 --> 00:15:16.399 power translates into the order one -third in 00:15:16.399 --> 00:15:18.320 Bessel world. Which leads to the Fourier transform, 00:15:18.580 --> 00:15:20.860 right? Right. If you take the Fourier transform 00:15:20.860 --> 00:15:23.559 of the Airy function... asking what frequencies 00:15:23.559 --> 00:15:26.500 make this up, you get a complex exponential with 00:15:26.500 --> 00:15:29.639 a cubic phase, e to the i key cubed. So in the 00:15:29.639 --> 00:15:32.580 frequency domain, it's a cubic. Yes. And that 00:15:32.580 --> 00:15:35.440 cubic phase is why it's so important in describing 00:15:35.440 --> 00:15:38.460 dispersion, how waves spread out. The Airy equation 00:15:38.460 --> 00:15:41.080 essentially describes a system with cubic dispersion. 00:15:41.139 --> 00:15:43.039 And there's this thing called the Stokes phenomenon. 00:15:43.759 --> 00:15:46.159 It sounds like a ghost story. It can be a nightmare 00:15:46.159 --> 00:15:49.019 for mathematicians. We said earlier that AI of 00:15:49.019 --> 00:15:52.139 X decays exponentially when X is positive, but 00:15:52.139 --> 00:15:54.500 that's only strictly true along the real axis. 00:15:54.820 --> 00:15:57.340 Meaning, if we stick to normal numbers. Right. 00:15:57.399 --> 00:16:00.000 But if you step off into the complex plane using 00:16:00.000 --> 00:16:02.759 imaginary numbers, the behavior gets complicated. 00:16:03.139 --> 00:16:05.120 Depending on the angle you look at the function 00:16:05.120 --> 00:16:08.200 in the complex plane, it might switch from decaying 00:16:08.200 --> 00:16:10.580 to growing. Wait, so if I just rotate my view 00:16:10.580 --> 00:16:13.259 in the complex plane, the fizzle becomes an explosion? 00:16:13.840 --> 00:16:16.539 Exactly. These sectors where the behavior flips 00:16:16.539 --> 00:16:19.559 are separated by lines called Stokes lines. It 00:16:19.559 --> 00:16:21.480 means you have to be very careful when you use 00:16:21.480 --> 00:16:24.120 approximations because the rules change depending 00:16:24.120 --> 00:16:26.779 on your angle of attack in the complex plane. 00:16:26.980 --> 00:16:29.899 Like a hall of mirrors. Very much so. But that 00:16:29.899 --> 00:16:33.440 complexity gives it its power. It allows it to 00:16:33.440 --> 00:16:36.059 connect different physical regimes. So what does 00:16:36.059 --> 00:16:38.299 this all mean for us? We've got an equation, 00:16:38.539 --> 00:16:42.409 y double prime minus xy equals zero. We've got 00:16:42.409 --> 00:16:45.009 rainbows, quantum tunnels, curving lasers, and 00:16:45.009 --> 00:16:47.309 random matrices. I think the lesson is about 00:16:47.309 --> 00:16:49.830 the universality of mathematics. Airy wasn't 00:16:49.830 --> 00:16:52.649 thinking about semiconductors or lasers in 1838. 00:16:52.850 --> 00:16:55.450 He was looking at a drop of water. But because 00:16:55.450 --> 00:16:57.750 he rigorously described the geometry of that 00:16:57.750 --> 00:17:00.230 turning point, he created a tool that unlocks 00:17:00.230 --> 00:17:02.669 the physics of boundaries across the entire universe. 00:17:03.049 --> 00:17:05.650 It's the tool for soft boundaries. Exactly. Nature 00:17:05.650 --> 00:17:09.289 rarely deals in hard, instantaneous stops. Things 00:17:09.289 --> 00:17:11.890 fade. Things transition. And the airy function 00:17:11.890 --> 00:17:14.769 is the map for that transition. I love that. 00:17:14.829 --> 00:17:16.490 So next time you see a rainbow, look for those 00:17:16.490 --> 00:17:19.049 faint extra bands on the inside. The supernumerary 00:17:19.049 --> 00:17:22.589 bands. And realize you are looking at the airy 00:17:22.589 --> 00:17:25.329 function in the wild. And think about that accelerating 00:17:25.329 --> 00:17:28.490 beam of light. It's a lovely image to leave on 00:17:28.490 --> 00:17:31.450 light, bending around a corner, guided by math 00:17:31.450 --> 00:17:34.309 from the 1800s. It really is. Thanks for diving 00:17:34.309 --> 00:17:35.910 in with us. We'll see you in the next one. Goodbye.