OK, so I want you to picture something for me
before we get started today. You are back in
your high school calculus class. Oh, boy. Right.
Maybe you loved it. Maybe you completely hated
it. You definitely remember the vibe. You've
got a piece of graph paper, a really sharp pencil,
and you draw a curve. And it's smooth. clean.
You can take a ruler, slide it right along that
curve, and at any single point, you can draw
a straight line that perfectly touches it. You
can find the slope. Right, the tangent line.
Exactly, the tangent line. And that right there
is the world, according to Isaac Newton. It's
this world of sliding down perfectly frictionless
slopes or planets orbiting in flawless ellipses,
apples falling in a straight line. It's elegant,
it is incredibly predictable, and above all,
it's smooth. It is. It's a really beautiful world.
I mean, it's the mathematics of the ideal. But
that actually brings us to the problem and really
the whole heart of our deep dive today. Because
that world, it's a lie. A lie. I mean, that's
a bit harsh for Newton, isn't it? Well, OK. Maybe
a useful fiction is a better way to put it. Because
the real world, the one you and I actually live
in, it isn't smooth. It's rough. It's porous,
it's jagged. And when you try to take that beautiful
smooth Newtonian calculus and apply it to something
messy, like. like water seeping through a cracked
rock, or the chaotic turbulence inside a jet
engine. Or even the stock market, right? Exactly.
The map doesn't just get difficult in those cases,
it fundamentally breaks. It just completely fails.
And that is exactly where today's topic comes
in. We are diving deep into a fascinating stack
of research on something called the fractal derivative.
Sometimes you'll see it called the Hausdorff
derivative in the literature. And honestly, looking
at these sources today, papers from researchers
like Wen Chen and Abdana Tangana, it really feels
like we're rewriting the rule book of physics
from the ground up. In many ways we really are.
We're looking at a completely new mathematical
framework, one that's designed to measure change
when the environment itself is a fractal. So
we're going to be getting into anomalous diffusion.
rethinking the very definition of velocity and
looking at some wild cutting edge work in what's
called fractal fractional calculus. So, yeah,
if you've ever wondered how scientists actually
model things that just refuse to follow the straight
and narrow path, things that get stuck, bounce
around, behave chaotically, this is the deep
dive for you. Exactly. We're basically asking
how do we measure speed? when the road is completely
full of potholes. I love that analogy. Let's
start right there with the failure of standard
physics, because I think that's the absolute
best hook for this. You mentioned that the old
rules break. Why? What is so inherently special
about a porous media or an aquifer that just
makes Isaac Newton's math give up and go home?
It really comes down to the assumptions we make
about space itself. Think about classical diffusion.
This is based on fundamental things like Fick's
laws or Darcy's law or Fourier's law for heat.
The classics. Right. And all of these laws are
built on the assumption of a random walk in free
space. OK, the random walk. I have heard this
term. It's usually described as a drunk person
stumbling around. Yeah, that is the classic analogy
in physics. Imagine a drunk person in a massive,
completely empty parking lot. OK. They take a
step, stumble in a totally random direction,
take another step. Over time, even though their
specific path is random, You can statistically
predict exactly how far they'll be from the center.
It follows a nice predictable bell curve, a standard
Gaussian distribution. Right, because the parking
lot is empty. There is absolutely nothing stopping
them. The space is smooth. Precisely. But now
takes that exact same drunk person and drop them
into a hedge maze. Oh wow. Or a jagged underground
cave system. or a sponge. Okay, now they're just
slamming into walls. They're hitting dead ends.
They're getting trapped in little feedback loops.
They have to retrace their steps. In the literature,
this is actually often called the ant in the
labyrinth problem. The ant in the labyrinth.
I really like that. Yeah, so if you're an ant
inside a complex porous rock, the actual distance
you travel isn't a straight line. It's this jagged
self -similar fractal path. you might have to
walk 10 meters just to move one meter of actual
displacement. Right. So if you try to use standard
calculus, which just implicitly assumes smooth
space and straight lines, your predictions will
be completely, laughably wrong. Because the speedometer
is basically broken since the road itself is
broken? That's a perfect way to put it. To fix
this, mathematicians eventually realized they
couldn't just... They couldn't just tack on a
correction factor to the old equations. They
had to redefine the very core concepts of distance
and velocity. They had to transform the actual
scales of space and time. And this is where it
gets really mind -bending to me. In the source
material, it talks about transforming these scales.
Instead of just looking at standard position
x and standard time t, we are suddenly looking
at x to the power of beta. and t to the power
of alpha. Yes. This is the fundamental ground
-up shift. We are officially entering fractal
spacetime. In the math, this is denoted as S
super alpha comma beta. S alpha. Right. And in
this specific space, the traditional definition
of velocity, which is just change in distance
divided by change in time, it makes absolutely
no sense because the space is non -differentiable.
Non -differentiable. Let me make sure I have
that. That basically means you cannot draw a
tangent line, right? Because no matter how far
you zoom in, it's still jagged. Exactly. Think
of a classic fractal like the coastline of Britain.
From space, it looks kind of curvy and smooth.
You zoom in and it's jagged rocks. Right. You
zoom in on a single rock. It's full of jagged
little pebbles and cracks. There is no magical
point where it suddenly becomes a perfectly smooth
line. So you literally cannot measure a traditional
slope. So how on earth do you measure speed in
a world where there are no smooth roads at all?
You have to redefine it entirely. In this new
fractal framework, velocity, let's just call
it v prime, is defined as the derivative of x
to the power of beta with respect to t to the
power of alpha. So dx to the beta divided by
dt to the alpha. That is just, it's so strange
to think about. You are not moving through standard
time anymore. You are moving through fractal
time. Essentially, yes. You're scaling the time
and the space to perfectly match the roughness
of the specific medium you're traveling through.
I like to think of it like currency exchange
rates. OK. How so? If you're traveling in a country
where the local currency is roughness, you can't
pay for things with standard smooth Newtonian
dollars. You have to convert them. Oh, I see.
T to the alpha and x to the beta. Those are your
converted fractal currency. OK, that makes a
lot of sense. We are fundamentally calibrating
our clock and our ruler to match the complexity
of the maze, but We definitely need to talk about
the derivative itself. I mean, the outline for
this deep dive is literally focused on measuring
the roughness, and the tool for that is the fractal
derivative. How do we actually calculate this
thing? I'm looking at the core definition here
in the papers, and it looks vaguely familiar
to what I learned in high school, but twisted.
It is familiar. That's honestly the beauty of
how they constructed it. In standard calculus,
the derivative is just a limit. It's the rise
over run as the run gets infinitely microscopically
small. You take f of t, 1 minus f of t, and you
divide it by t1 minus t. Right, just finding
the slope. Exactly. The fractal derivative rigorously
keeps that same basic structure, but it swaps
out the denominator. Instead of measuring the
change against linear time, which is that t1
minus t, we measure it against our new fractal
time. Okay, so the denominator becomes, looking
at the equation, it becomes t1 to the alpha minus
t to the alpha as t1 approaches t. So we are
comparing the change in whatever function we're
looking at, not to the ticking of a standard
wall clock, but to the ticking of this newly
scaled fractal clock. Exactly. And the really
deep mathematical motivation for this actually
comes from something called the Taylor series.
Oh man, the Taylor series. I vaguely remember
that being the way we approximate really complex
curves by just layering simple polynomials together.
That's the one. Let's use an analogy. Think about
early 3D video game graphics from the 90s. Oh,
yeah, super low poly count like the original
Lara Croft with a triangular nose. Right. Standard
calculus is basically like trying to render a
highly complex organic face using big flat triangles.
It's a functional approximation. We say any smooth
function can be approximated by a sum of straight
lines. But in the fractal world, straight lines
simply do not fit. They don't hug the roughness
of the reality. So the standard triangles are
just too big. Or, rather, they're entirely the
wrong shape to begin with. If you try to build
a perfectly round ball out of square Lego bricks,
it's always going to look blocky. But if you
have special curved Lego bricks, or in our case,
fractal bricks, you get a significantly better
fit. Right. So researchers generalized the standard
Taylor series. Instead of expanding with straight
lines, we approximate the function using terms
like x to the alpha minus x zero to the alpha.
So we are essentially upgrading our underlying
graphics engine. We're moving from PS1 straight
flat lines to a high definition fractal rendering
engine. That is a perfect analogy for what the
math is doing. And this specific upgrade leads
to what the source is called the fractal Maclaurin
series. It gives us a way to take a function
that has fractal support, meaning it literally
lives on a fractal set and expand it. And then
let's just treat it with the exact same level
of rigorous mathematical analysis as we treat
standard, smooth functions. We aren't just guessing
or throwing our hands up anymore. We have a highly
systematic toolkit to break down these rough
functions. Now, whenever we learn new math rules,
the absolute first question for anyone doing
the homework is always, Do the old shortcuts
still work? Like the chain rule. The chain rule
absolutely saved my life in calculus. Does that
survive the transition into the fractal world?
It does survive, but again with a very specific
twist. The papers show that if you want to take
the fractal derivative of a function f with respect
to x to the alpha, you can absolutely still use
the standard chain rule, but you have to multiply
it by a scaling factor. Okay, I see the formula
right here in the text. It says the fractal derivative
is equal to 1 over alpha times x to the 1 minus
alpha, all multiplied by the standard derivative.
Right, and really think about what that equation
means physically. It acts as a bridge. A bridge.
Yeah, it links the strange, jagged fractal derivative
directly back to the standard smooth derivative
we already know and love. It mathematically proves
that these aren't two completely isolated separate
universes. They are intimately connected by that
exact scaling factor. I see. It's like that exchange
rate we mentioned earlier. You can still use
your US dollars, your standard derivative, but
you have to pay the exchange fee, which is that
x to the 1 minus alpha term, to actually spend
them. in the fractal country. Exactly. And establishing
that mathematical bridge is incredibly crucial
if you actually want to apply this theory to
real life. You can't just stay in abstract pure
math land forever. Which perfectly brings us
to the killer app of fractal derivatives. The
whole reason this isn't just a fun abstract puzzle
for pure mathematicians to play with. Anomalous
diffusion. Anomalous diffusion. We've touched
on this a bit, like pollution seeping into an
aquifer or heat moving through a weird composite
material. But let's get really specific for a
second. Why does the standard model fundamentally
fail here? Well, the standard model, which is
usually governed by Fick's second law, assumes
that things always spread out at a normal, totally
predictable rate. But in porous media, you encounter
what statisticians call heavy tails. Heavy tails.
What does that look like? Imagine that standard
bell curve again. In a normal distribution, the
probability of finding a particle really far
away from the center drops off to zero incredibly
fast. But in anomalous diffusion, that bell curve
physically gets stretched out. Oh, okay. The
tails get heavier. Exactly. The sources heavily
discuss this linear anomalous transport diffusion
equation. It looks a lot like the standard diffusion
equation on the page, but instead of standard
derivatives, we plug our new fractal derivatives
right into it. Let me look at equation one from
the notes here. It says, The change in concentration,
u, over fractal time, t to the alpha, is equal
to a diffusion coefficient, d times the change
in space, x to the beta. Yes. And when you actually
sit down and solve that differential equation,
and this is truly where the magic happens, you
do not get a standard bell curve. You get what
they call a stretched Gaussian kernel. Trying
to visualize that. It's like someone grabbed
the far edges of the bell curve and just pulled
them violently outwards. Yes, or stepped on the
top and squashed it down. Statically, the fundamental
solution involves terms like e to the negative
x to the power of 2 beta, all divided by 4t to
the alpha. And that specific shape tells us something
incredibly vital. It tells us that a pollution
plume might travel much, much faster than standard
physics expects. That's super diffusion, because
maybe it found an underground highway and the
rock cracks. Or, conversely, it might travel
much slower, sub -diffusion, because it keeps
getting stuck in dead ends. And there is a specific
metric for quantifying the speed, right? I'm
looking at something called the mean squared
displacement. Yes. Mean squared displacement
is the absolute gold standard for measuring any
kind of diffusion. In normal smooth physics,
mean squared displacement simply grows linearly
with time. It's just proportional to two to the
power of one. Right. You wait twice as long,
the area covered doubles. Very simple. Exactly.
But here in the fractal world, the notes highlight
a very specific asymptote. The mean squared displacement
is proportional to time raised to a highly complex
power It's over to the power of 3 alpha minus
alpha beta all over 2 beta Wow 3 alpha minus
alpha beta divided by 2 beta. That is incredibly
specific. It is remarkably specific. And that
exact power law tells you precisely how anomalous
the diffusion is. If you physically measure your
alpha and beta parameters, which basically describe
the inherent fractal nature of your rock or your
specific turbulence, you can plug them in and
predict exactly how that pollution plume will
spread over 10 years. That is an incredibly powerful
tool for real world environmental engineering.
So what this all really boils down to is that
we can actually predict the chaos. We aren't
just looking at a cracked rock shrugging our
shoulders and saying, well, it's messy, so who
knows where the water goes. Exactly. It literally
turns chaos into a completely calculable problem.
But the story of this math actually doesn't end
there because fractal derivatives are, by their
very definition, completely local. They only
look at what's happening right here, right now,
at one specific point in the fractal space. But
the real world obviously has memory. What happened
yesterday affects today. It definitely does.
And that crucial realization leads us to the
absolute cutting edge of this field. Fractal
fractional calculus. Now this is the part of
the reading that sounded almost like science
fiction to me. We are now combining fractal derivatives
with fractional derivatives. And I have to just
stop you right here because these two words sound
almost entirely identical. Can you clearly distinguish
between fractal and fractional for us? Oh, it
is a hugely common point of confusion. Let's
break it down very simply. A fractal derivative,
as we've been discussing, deals entirely with
scaling the local measure. It deals strictly
with the geometric jaggedness of the space itself.
It basically answers the question, how physically
rough is the road right under my tires right
now? Okay, so I'm writing that down. Fractal
equals geometric roughness. Right. A fractional
derivative is entirely different. It is generally
a non -local operator. It involves taking an
integral antler over the entire history of a
system. It answers the question, where exactly
have you been before you arrived at this current
point? It fundamentally captures the memory of
the system. OK, so fractal is about the jaggedness
of the space, and fractional is about the memory
of the past. Beautifully put, yes. And fairly
recently, brilliant researchers like Professor
Abdana Tangana from South Africa have been working
tirelessly on actually merging these two massive
concepts together. They successfully introduced
what are called fractal -fractional differential
operators. This honestly sounds like the Avengers
of calculus. You are taking the space scalar
and teaming them up with the history keeper.
That's a surprisingly accurate way to visualize
it. But Tangana and other mathematicians realize
that highly complex real -world systems almost
always have both fractal geometry and intense
memory effects happening simultaneously. So the
rigorously created operators that combine them.
And the source material lists three very specific
ways. three distinct kernels to actually compute
this. Right, I saw these in the text. The power
law kernel, the exponentially decaying kernel,
and the generalized Mideg -Leffler kernel. That
last one sounds like an absolute heavy hitter.
Oh, it really is. The Mideg -Leffler function
is essentially considered the queen of fractional
calculus. But the underlying reason we even have
three different kernels is what's truly fascinating.
It's because mathematical memory isn't all exactly
the same. How so? Like, does math seriously have
different distinct types of amnesia? In a very
real way, yes. Think about how human beings forget
things. I try hard not to. Well, let's look at
the power law, Colonel. It describes a system
with what we call long memory. The effects of
the past do decay over time. But they decay very,
very slowly. They stick around for ages. This
is extremely common when you're modeling things
on vast geological time scales. Okay, so it's
like a grudge that just never quite goes away.
Exactly. Now, compare that to the exponentially
decaying kernel. That one describes a memory
that feeds away relatively fast. It's essentially
short -term memory. What happened 10 minutes
ago matters a lot to the system, but what happened
yesterday is just effectively gone. And then
there's the mid -egg -leffler kernel. Right.
The generalized mid -egg -leffler kernel. That
describe a highly complex non -local crossover
between multiple different behaviors. A system
might actually start with a really fast memory
decay and then dynamically shift to a slow one
later on. It is by far the most sophisticated,
nuanced tool in the entire box. So depending
entirely on whether you are trying to model a
slowly drying aquifer or a rapidly spreading
virus or an unpredictable stock market crash,
the actual memory of the system functions differently
and you just pick the specific kernel that fits
that reality. Exactly. It's absolutely not a
one -size -fits -all situation. It gives scientists
a deeply customizable toolkit to describe the
precise nuance of decay and memory inside a very
rough fractal environment. I really want to touch
on this non -local concept just a bit more before
we wrap up. The deep dive outline explicitly
mentions fractal non -local calculus, and it
talks about integrals over a specific range.
Yes. This directly ties back to the idea that
in these complex systems, the current state of
a single point depends heavily on its surrounding
neighborhood. The strict mathematical definitions
involve taking integrals over a range, usually
denoted as a tecus or x to b. And this is honestly
where the math gets intensely geometric. because
we are only integrating over what's called the
perfect fractal set, f to the alpha. Oh, right.
And the derivative is mathematically defined
differently depending on whether you are actually
physically inside that fractal set or in the
empty gaps between it. Correct. The source text
explicitly mentions that the derivative is rigorously
defined via a limit if your point T is actually
inside the set F. But if E is outside the set,
meaning it's in the empty void space of the sponge,
so to speak, the derivative is just mathematically
zero. That is just wild. It effectively treats
the empty physical space as if it straight up
doesn't exist. It absolutely treats the fractal
structure as the only reality that matters. The
rough network itself is the entire universe.
Okay, let's take a deep breath and step back
for a second. We have covered a massive amount
of ground today. We essentially started with
a sponge or porous media and realized that our
standard trusted speedometers and standard clocks
are dx and dt. They just flat out do not work
in there. They give us completely false, useless
readings. So mathematics had to invent a totally
new clock, t to the alpha, and a totally new
ruler, x to the beta. We completely rebuilt the
Taylor series from the ground up to fit this
new world, upgrading our graphics engine, as
we joked earlier. And we found out that doing
all of this allows us to finally model how things
like pollution practically spread in an aquifer,
using these weird stretched Gaussian curves instead
of standard bell curves. Which is absolutely
vital for public safety and actual civil engineering.
And then, just to push it over the edge, we realized
that we can physically combine the space scaling
math with fractional calculus to completely account
for a complex system's memory using intense tools
like the Mitakag -Luffler kernel to decide if
a system has a long -term grudge or just short
-term memory. That is an absolutely excellent
summary of the entire framework. We've literally
gone from making a basic observation that rocks
are rough to developing a highly sophisticated,
robust mathematical language fractal fractional
calculus that can accurately describe the complex
flow of energy and matter straight through that
roughness. It's just so impressive to me because
it proves that math isn't just a static dead
thing. We always kind of think of calculus as
this immutable stone tablet that was handed down
by Newton and Leibniz centuries ago, but it's
actually actively evolving right now to meet
the sheer complexity of the real world we keep
discovering. That is honestly the absolute most
important takeaway for me from all these sources.
The fractal derivative is not just some abstract
mental gymnastics for academics. It is a strict
necessary evolution. If we ever want to truly
understand the real world, which, let's face
it, is rarely ever smooth and simple, we absolutely
need mathematics that speaks the native language
of roughness. We have to be able to model chaos
and turbulence and anomalous diffusion accurately,
or else our bridges fall down and our climate
models completely fail. It effectively gives
us the lens to finally see the underlying order
hidden deep inside the rough edges. It does.
It really does. And, you know, if I can just
leave you with one final thought that pushes
this entire concept just a little bit. Oh, please
do. This is where it always gets really interesting.
So the sources mention a specific paper by Wen
Chen, and the title is, Time -Space Fabric Underlying
Anomalous Diffusion. It actively suggests that
the mere existence of anomalous diffusion implies
that there is an underlying fractal time -space
fabric to reality. A fractal time -space fabric.
Really think about the implications of that.
We universally assume that time just flows exactly
the same for everyone everywhere at a perfectly
constant, smooth rate. Right, time is time. But
in these incredibly complex, porous systems,
the physical math works best if we fundamentally
assume that time itself actually scales differently
depending on the specific fractal dimension of
the space. Whoa! If time literally scales as
t to the alpha when you're inside a turbulent
cloud, What does that imply about our basic human
perception of flow and causality and change when
we look at complex systems? So in a very real
mathematical way, the actual clock of the universe
might tick differently depending entirely on
just how inherently rough the neighborhood is.
It's a deeply provocative thought. The math definitely
points in that direction. Well, that is definitely
something for all of you to just stare at the
ceiling and think about tonight. Thank you so
much for guiding us through the incredibly jagged
complex world of the fractal derivative today.
It was my absolute pleasure. It's always a really
fun time exploring the rough edges of reality.
And a huge thank you to everyone listening for
taking the plunge into the math of the rough
and jagged with us. We will catch you on the
next deep dive.