What is a random walk?

Welcome to Mass Is Not Boring by Kidopoly.com. I'm Mira. And I'm Finn. If you haven't already, tap follow wherever you're listening so you never miss an episode.

Okay, what ordinary thing is going to look completely different today?

Imagine an enormous, endless grid of paths, like the biggest sheet of square graph paper you have ever seen, stretching on forever.

Okay, I'm picturing an endless grid. Like city streets but perfectly square.

Right in the middle of this giant grid is a tiny ant. At every crossing, she rolls a four-sided dice to decide where to go. Left, right, forwards, or backwards.

Completely by chance. She doesn't have a plan.

No plan at all. And she does this over and over, taking millions of steps in totally random directions.

She is definitely getting hopelessly lost. There is no way she ever finds her way home.

That's exactly what you would think. But if she keeps wandering on that flat grid, she is mathematically guaranteed to step back on the exact spot she started.

Wait, guaranteed? But you just said she is choosing completely at random. How can pure chance guarantee anything?

That is the magic of what mathematicians call a random walk. And to find the first person who really noticed it, we have to travel back to London in the year 1827. 1827? Were they looking at ants? No. We are visiting a famous botanist named Robert Brown. He was looking through his microscope at something much smaller.

Pollen. From a pink flower called Clarkia pulchella.

Okay, so just yellow flower dust. What's so weird about that?

Well, when the pollen was sitting in water, it swelled up and popped open. And Brown noticed there were even tinier specks inside. Particles from inside the pollen grains.

Wait, I thought pollen was already just a tiny speck of dust. You're telling me there is smaller stuff inside the speck?

Exactly. But that wasn't the weird part. When Robert Brown looked at those tiny specks floating in the water, they were doing this incredibly strange, jittery dance. Like they were swimming? Not swimming like a fish with a destination. They were darting left, then right, then zigzagging backwards, totally aimless, jiggling around completely at random.

Were they alive? Like incredibly tiny bugs. That is exactly what he thought at first. So he tried looking at dead pollen. They still jiggled. He tried crushing up old dry rocks and looking at the dust in water. The rock dust jiggled the exact same way.

So they weren't alive. But things don't just dance by themselves. What was pushing them?

For almost 80 years, that jittering dance was a complete mystery. Scientists called it Brownian Motion after Robert Brown, but nobody knew why it happened.

So who finally cracked the code?

We have to fast forward to the year 1905, to the city of Bern in Switzerland. A 26-year-old scientist working in a patent office finally figured it out.

Wait, 1905, Swiss Patent Office. Is this Albert Einstein? It is Albert Einstein. The famous wild hair guy solved the dancing dust puzzle.

Yes, Einstein realised the invisible water wasn't just smooth liquid. It is made of billions and billions of individual invisible water molecules. And those molecules are always zipping around, completely at random.

Like a massive invisible bumper car ride.

Exactly. Imagine a gigantic beach ball floating in a crowd of thousands of people moving randomly. They are going to bump into the beach ball from all different sides, pushing it left, then right, then backwards.

Oh, so the tiny specks from the pollen are the beach balls, and the invisible water molecules are smashing into them over and over.

You got it! Einstein proved that the specks were taking a random walk, forced by the millions of tiny, random crashes from the invisible water molecules.

Okay, that is super cool. But what does the jiggling pollen dust have to do with the ant from the beginning? The one you said was guaranteed to get back home?

That brings us to our final stop. 1921, in Zurich, Switzerland. A Hungarian mathematician named George Pollier was thinking deeply about these random walks. What was his big question? He wondered, if a random wanderer, taking step after step by pure chance on a grid, keeps walking forever, do they always find their way back to where they started?

And you said yes. But I still don't believe it. I feel like you just wander away to the edge of the grid.

That's what George Polly's math showed. On a flat two-dimensional grid, a random walk returns to its starting point with probability one. That means, given unlimited time, it always gets home. Always.

So the ant rolling the four-sided dice is completely safe. She will definitely cross her starting point again.

Yes, but here is where the story turns on its head. Pollya asked another question. What if the wanderer isn't stuck on a flat piece of paper? What if the wanderer is a bird?

A bird. So it can fly forwards, backwards, left, right, and also up and down?

Yes, a three-dimensional random walk. Now the bird is rolling a six-sided dice to choose its next direction. Does the bird always get home? The answer is no. A wanderer that can move up and down, like a bird or a fish, gets home only about one time in three, just 34% of the time.

Wait, what? Just by adding up and down, it drops from guaranteed to mostly lost.

Yes. The extra space of that third dimension makes the universe so much bigger and so much emptier. A random wanderer in three dimensions will most likely drift away forever.

That is a little bit lonely. The bird is just lost in the big open sky.

But it's also incredibly beautiful. Because this random walk rule is what makes the world work. Think about what happens when you drop a single drop of blue ink into a glass of still water.

Oh, the ink molecules get smashed by the water molecules, just like the pollen specs.

Exactly. And because the water is three-dimensional, the ink molecules drift away in a three-dimensional random walk. They almost never return to the centre.

Which is why the whole glass of water turns blue, instead of the ink staying in one single dot.

Yes, it's exactly the same sideways connection when you smell perfume from across the room, or when smoke spreads in the air. George Pollier showed us that even pure chaotic chance has a shape. Alright, Finn, time for the quiz. Are you ready?

Okay, I'm ready. Let's see if I remember everything.

First question. What is the real mathematical name for this idea? Next one. What year did botanist Robert Brown notice the tiny specks jiggling in water? Question 3. Which famous scientist figured out the specks were being hit by invisible molecules in 1905? Here's number four. What happens to an ant on a flat two-dimensional grid if she walks randomly for an unlimited time?

Last question. Does a three-dimensional random wanderer, like a bird or a fish, always get home? Great job! Let's see how you did. The first answer is it is called a random walk. For question two, it was the year 1827. The answer to question three is Albert Einstein. For question four, she will always return to her exact starting point.

And the final answer is no, they only get home about one in three times and might drift away forever.

I am never going to look at a drop of ink the same way again. It's just a billion tiny random walks drifting away. It is.

And the idea that random chance is the reason a drop of ink spreads out is just so cool.

I'm definitely going to think about the lost bird every time I look at the sky, too.

If you loved learning about random walks today, please leave us a five-star review. Just scroll down on your app and tack the stars. It helps more people find these mathematical mysteries.

Oh, and we have sister shows too. History, science, and philosophy is not boring. Go give those a listen and a follow as well.

And if you want a shout out on the show, or just want to say hi, email us at hello at kidopoly.com.

See you next time on Maths Is Not Boring. Bye!

What is a random walk?

Kapitelklicken springt direkt hin

Shownotes

Transkriptvom Podcast bereitgestellt · Zeitmarken springen hin