Welcome back to the Deep Dive. Today we are standing
on the edge. Ideally, not literally. No, not
literally, but metaphorically. We're talking
about boundaries. You know that moment where
a system just shifts? Right. Like on one side,
everything is vibrating and energetic, moving
like a wave, and then you cross this invisible
line and suddenly everything just changes. It
gets quiet. It decays. That is a very poetic
way to describe what's called a turning point.
I try. But seriously, we are looking at a specific
mathematical concept that describes exactly that
transition. It's called the airy function. And
I know you just heard function and might be reaching
for the skip button, but don't do it. Because
this isn't just dry calculus. This is the math
of rainbows. It is. And it's the math of quantum
mechanics and these strange beams of light that
seem to curve all on their own. It is a surprisingly
beautiful bridge between two very different worlds
of physics. So we have a stack of notes here,
primarily based on the Wikipedia entry for the
Airy function, which actually goes into some
serious depth on the history and the heavy math.
Our mission today is to decode this. We want
to understand how one little equation acts as
this bridge you mentioned. So let's start with
the name, Airy. Is it named that because it describes
air? It's actually named after a person, Sir
George Bedell Airy. Oh, okay. He was the British
Astronomer Royal back in the 19th century. Okay.
Astronomer Royal. That's a cool title. It is.
And like a lot of scientists of his time, he
wasn't just looking at stars. He was obsessed
with optics, how light behaves. Before we get
to the history, we really have to look at the
beast itself, the equation. Right. The Airy equation
or the Stokes equation, as it's sometimes called.
I'm looking at it here and it looks deceptively
simple. It's just y double prime minus xy equals
zero. Exactly. yxy equals zero. Exactly. yxy
equals zero. Okay, let's unpack this. For anyone
who hasn't done calculus in a while, y double
prime, that's the second derivative, that describes
curvature, right? Or acceleration. Correct. In
physics, the second derivative tells you how
the function is bending. If you think of a graph,
it's the concavity. So the equation says that
this curvature is equal to x times the value
of the function itself. And that simple relationship
creates this whole turning point behavior. It
does. And this is why mathematicians love it.
It is considered the simplest second -order linear
differential equation that has a turning point.
But to see why, you have to sort of play with
the variable x. Okay, I'm game. Let's say x is
a negative number. We're on the left side of
the graph. If x is negative, say, minus 5, then
the equation becomes y. double prime equals negative
five y or y double prime plus five y equals zero
wait i recognize that structure acceleration
equals negative position that's that's a spring
exactly it's the equation for a simple harmonic
oscillator a spring bouncing back and forth,
or a wave, the curvature is always pulling the
function back toward zero. So when you look at
the graph of the airy function on the left side,
where x is negative, it's wiggly. It oscillates.
Precisely. But notice what happens as you go
further left into more negative numbers. The
spring gets stiffer, you could say, because x
is a bigger negative number. So the waves get
faster. Yes. The frequency increases, and the
amplitude, the height of the waves, gets smaller.
It's a nervous, fast ripple. Okay, so on the
left, we have ripples. Now, let's cross that
line. What happens when we hit zero, when x becomes
positive? That is the turning point. At x equals
zero, the sign flips. Now you have y double prime
equals xy, where x is positive. So the curvature
isn't pulling it back to zero anymore. It's pushing
it away. Exactly. If the function is positive,
the curvature bends it more positive. If it's
negative, it bends it more negative. It stops
oscillating. It becomes an exponential function.
And exponential usually means it just takes off,
right, or crashes. In the case of the main area
function, which we call AI of X, it crashes.
It decays rapidly. It just slides down towards
zero and fades away. So if you were to draw this,
on the left you have these ripples, and then
they hit this invisible wall at zero, and the
ripples just smooth out and die off into a long
tail on the right. That is the perfect visualization.
And that is why it's so important in physics.
Because in the real world... Boundaries are rarely
hard. If a wave hits a wall, it doesn't just
stop instantly at the atomic level. It penetrates
a little bit. It decays. The Airy function describes
that soft boundary where oscillation turns into
decay. That is fascinating. It's like the mathematical
description of fizzling out. In a way, yes. But
here is where it gets really interesting. Differential
equations like this usually have two solutions.
We call them linearly independent solutions.
Right. I saw this in the notes. We have AI of
X, which is the good one we just talked about.
But then there's an evil twin. I wouldn't call
it evil, but it is certainly unruly. The first
solution is AI of X. It's well behaved because
as X gets bigger, it goes to zero. That makes
it physically useful. For things like probability
or particles trapped in a box. Exactly. And the
second one? The second one is called B of X.
B -I. B -I. Okay, what does B do? On the negative
side, the wave side, B looks just like A -I.
It oscillates with the same amplitude, just shifted
in phase a bit. But on the positive side... where
AI politely fades to zero, B explodes. Explodes.
It grows exponentially to infinity. Oh, wow.
So if AI is the fizzle, B is the kaboom. Basically.
And because it shoots off to infinity, it's often
discarded in simple physical problems where you
need a finite answer. You can't have a probability
of finding a particle that is infinity, you know?
So why do we keep it around? Why not just delete
it from the textbooks? Because mathematically,
you need it to form a basis. You can't define
the general solution to the differential equation
without it. And sometimes you are describing
a system that grows. So B is the necessary partner,
even if he's a bit loud. Got it. The unruly sibling
stays in the picture. Now let's pivot back to
the man himself, George Bedell Airy. You said
he didn't invent this just to torture calculus
students. He was looking at rainbows. He was.
And not just the cartoon Rainbow. He was studying
something called the supernumerary rainbow. Supernumerary.
That sounds expensive. It does. Have you ever
looked at a really bright rainbow and noticed
that just inside the main violet band there are
these faint repeating pink and green bands? I
think so, like echoes of the rainbow. Exactly,
those are supernumerary rainbows. Classical geometric
optics just drawing lines for light rays couldn't
explain them. According to simple ray theory,
the light should just stop at the edge of the
rainbow. But it doesn't. No, light is a wave
and waves interfere with each other. Ari was
trying to calculate the intensity of light near
a caustic. Define caustic for us, because I usually
think of caustic wit or acid. In optics, a caustic
is an envelope of light rays. It's where light
rays bunch up and focus. You know when you shine
a light through a glass of water and you see
that bright curved shape on the table? Yeah,
that sharp curve of light. That's a caustic.
A rainbow is essentially a giant caustic in the
sky. It's a fold in the light intensity. Airy
realized that near this edge, the intensity of
the light doesn't just cut off. It oscillates
and then fades. Wait, oscillates and then fades?
That sounds familiar. It is exactly the Airy
function. He developed this math specifically
to describe how the brightness of the rainbow
ripples near that edge. That is so cool. So he
writes this down in... When was this? 1838. 1838.
He writes down the math. Was he right? He was.
A few years later, in 1841, a physicist named
William Hallows Miller tested it. He didn't have
a convenient rainstorm, so he used a thin cylinder
of water to simulate a raindrop and shone light
through it. He observed 30 of these tiny dark
and bright bands, these supernumerary bands,
and they matched Airy's theory perfectly. I love
that. 30 bands. That's precision. It is. And
interestingly, Airy didn't call it the Airy function
at the time. The notation AI of X was actually
introduced much later by a geophysicist named
Harold Jeffries, but the name stuck to Airy.
As it should. Okay, so we have rainbows. That's
a... Beautiful classical application. But the
outline says this function is a heavy hitter
in quantum mechanics, too. Oh, absolutely. In
fact, for a modern physics student, the Airy
function is almost synonymous with quantum mechanics.
Why? What's the connection? Think back to the
shape of the function. Ripples on one side, decay
on the other. In quantum mechanics, particles
behave like waves. Right. Wave -particle duality.
So imagine a particle trapped in a potential
well. Specifically, a triangular well. Imagine
a floor that is sloped, like a ramp, and a hard
wall on one side. Okay, so the particle can bounce
against the wall, but as it goes up the ramp,
a force slows it down until it stops and slides
back. Exactly. Classically, there is a specific
point on the ramp where the particle stops and
turns around. A turning point. There's that phrase
again. In quantum mechanics, the particle is
a wave. On the ramp side, before the turning
point, the particle has energy, so its wave function
oscillates. That's the Wigley part of AO of X.
But past the turning point, where classically
it shouldn't be able to go. It tunnels. It tunnels.
It penetrates into the forbidden region. And
that penetration is perfectly described by the
decay tail of the Airy function. So the Airy
function is literally the map of a particle hitting
a soft wall. Exactly. It solves the Schrodinger
equation for a particle in a constant force field
like a constant electric field. It is the textbook
description of a boundary in quantum mechanics.
This brings me to something else in the notes.
The WKB approximation. The notes call it a uniform
semi -classical approximation. That is a mouthful.
What does it actually mean? This is where the
Airy function does some heavy lifting. In physics,
we often can't solve equations perfectly. We
use approximations. The WKB method is a way to
approximate the wave function when the properties
of the environment are changing slowly. Okay,
so it's a shortcut. Yes, and the shortcut works
great when the particle is moving fast, far from
the turning point, or when it's deep in the forbidden
region. But right at the turning point, the approximation
breaks. Why does it break? Because the approximation
relies on the momentum of the particle. At the
turning point, the velocity goes to zero. The
wavelength effectively tries to become infinite.
The math just, it blows up. You get a divide
by zero error. So the shortcut fails exactly
where the action is happening. Exactly. So physicists
use the Airy function as a patch. They know that
locally... Right near the turning point, the
system looks like the area equation. So they
solve that little section using AI of X and then
glue it to the WKB solutions on either side.
It's the duct tape of quantum mechanics. A very
elegant, high -precision duct tape, yes. It connects
the oscillating world to the decaying world smoothly
without the math blowing up. I love the idea
of a patching function. Now, I want to talk about
one application that jumped out at me from the
notes. This accelerating beam in optics. This
sounds like sci -fi. This is one of the coolest
modern applications. It was theoretically proposed
and then realized in the lab around 2007. Imagine
a laser beam. Usually light travels in a straight
line, right? Rule number one of light. Light
travels straight. Well, scientists created what
they call a transversely asymmetric optical beam.
Basically, they shaped the cross section of the
beam so that the intensity profile looks exactly
like the airy function. So a bright peak on one
side and then those fading ripples on the other.
Exactly. When you launch this beam, something
bizarre happens. The main brightest peak appears
to curve. It accelerates to the side. It curves
like a banana cake in soccer. Yes. It creates
a parabolic arc. as it travels through free space.
But physics, conservation of momentum, you can't
just have light turn left for no reason. It needs
to bounce off something. Physics is safe, don't
worry. This is where the tail of the airy function
comes in. Remember that long tail of ripples
on the other side? The part that fades out. Yes.
As the main break peak curves one way, that low
intensity tail spreads out and curves in the
opposite direction. Oh, I see. So the average
position of the light is still going straight.
Exactly. The center of gravity of the beam travels
in a straight line, satisfying conservation of
momentum. But visually, and in terms of where
the energy is focused, the beam effectively bends
around corners. That is wild. So you could shoot
a laser beam around an obstacle? In principle,
yes. They are called airy beams. They are also
non -diffracting, meaning they don't spread out
and get fuzzy as fast as normal laser beams.
Just because they used this specific math shape.
Because the airy function has these special properties
of stability. And get this, it's self -healing.
Self -healing. Like Wolverine. A little bit.
If you place a small obstacle in the path and
block the main bright peak, the tail, because
it contains the phase information of the whole
function, will actually reconstruct the main
peak further down the line. No way. The tail
grows new head. It does. The interference pattern
reforms the airy shape. Okay, my mind is officially
blown. We went from a 19th century astronomer
looking at rainbows to self -healing curving
laser beams that fix themselves if you block
them. And we aren't even done. It shows up in
probability theory too. Right, the random matrices.
This seems like a totally different field. We're
leaving physics now and entering statistics.
We are. Random matrix theory deals with the statistics
of huge grids filled with randomness. numbers.
It turns out, if you look at the distribution
of the largest eigenvalues, the most important
numbers describing these matrices, they follow
a specific pattern. Let me guess. It's airy.
It's the Tracy -Whidham distribution, which is
defined using the airy function. Why airy? Where
is the turning point in a random grid of numbers?
It represents the soft edge of the spectrum.
The eigenvalues are packed together like a fluid,
and at the edge of that fluid, the density drops
off. That drop -off is described by the Airy
function. So it's a boundary again. It's always
a boundary. This pops up in everything from nuclear
physics to the behavior of the stock market to
how distinct interface growth happens, like how
a coffee stain spreads or how a Tetris game fills
up. The Airy function describes Tetris. In a
very abstract statistical way via the KPZ equation,
yes, it describes the fluctuations of the surface
growth. If you drop blocks randomly, the roughness
of the top layer follows statistics governed
by the Airy function. It really is the boundary
function. Wherever there is an edge, Airy is
there. That is the big takeaway. Before we wrap
up, I do want to dip our toes into the deep part
of the notes, the mathematical nuances. We talked
about AI and buy, but the notes mentioned Bessel
functions. Are they related? They are cousins.
The Airy functions can actually be written in
terms of Bessel functions of fractional order,
specifically order one -third. Why one -third?
Is that arbitrary? No, it comes from the cubic
nature of the phase. Remember the integral definition
of the Airy function. It involves t cubed over
three. That cubic term is the signature of the
Airy function. When you do the math, that cubic
power translates into the order one -third in
Bessel world. Which leads to the Fourier transform,
right? Right. If you take the Fourier transform
of the Airy function... asking what frequencies
make this up, you get a complex exponential with
a cubic phase, e to the i key cubed. So in the
frequency domain, it's a cubic. Yes. And that
cubic phase is why it's so important in describing
dispersion, how waves spread out. The Airy equation
essentially describes a system with cubic dispersion.
And there's this thing called the Stokes phenomenon.
It sounds like a ghost story. It can be a nightmare
for mathematicians. We said earlier that AI of
X decays exponentially when X is positive, but
that's only strictly true along the real axis.
Meaning, if we stick to normal numbers. Right.
But if you step off into the complex plane using
imaginary numbers, the behavior gets complicated.
Depending on the angle you look at the function
in the complex plane, it might switch from decaying
to growing. Wait, so if I just rotate my view
in the complex plane, the fizzle becomes an explosion?
Exactly. These sectors where the behavior flips
are separated by lines called Stokes lines. It
means you have to be very careful when you use
approximations because the rules change depending
on your angle of attack in the complex plane.
Like a hall of mirrors. Very much so. But that
complexity gives it its power. It allows it to
connect different physical regimes. So what does
this all mean for us? We've got an equation,
y double prime minus xy equals zero. We've got
rainbows, quantum tunnels, curving lasers, and
random matrices. I think the lesson is about
the universality of mathematics. Airy wasn't
thinking about semiconductors or lasers in 1838.
He was looking at a drop of water. But because
he rigorously described the geometry of that
turning point, he created a tool that unlocks
the physics of boundaries across the entire universe.
It's the tool for soft boundaries. Exactly. Nature
rarely deals in hard, instantaneous stops. Things
fade. Things transition. And the airy function
is the map for that transition. I love that.
So next time you see a rainbow, look for those
faint extra bands on the inside. The supernumerary
bands. And realize you are looking at the airy
function in the wild. And think about that accelerating
beam of light. It's a lovely image to leave on
light, bending around a corner, guided by math
from the 1800s. It really is. Thanks for diving
in with us. We'll see you in the next one. Goodbye.