シャドーイング練習: You've (Likely) Been Playing The Game Of Life Wrong - 動画で英語スピーキングを学ぶ

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Some things are not normal.

By that I mean if you go out in the world and start measuring things like human height, IQ, or the size of apples on a tree,

you will find that for each of these things most of the data clusters around some average value.

This is so common that we call it the normal distribution.

But some things in life are not like this.

Nature shows power laws all over the place.

That seems weird.

Like, is nature tuning itself to criticality?

If you make a crude measure of how big is the world war by how many people it kills, you find that it follows a power law.

The outcome will vary in size over 10 million, 100 million.

It's much more likelihood of really big events than you would expect from a normal distribution.

And they will totally skew the average.

The system you're looking at doesn't have any inherent physical scale.

It's really hard to know what's going to happen next.

The more you measure, the bigger the average is, which is really weird.

It sounds impossible.

It's very important to try to understand, you know, which game you're playing and what are the payoffs going to be in the long run.

In the late 1800s, Italian engineer Vilfredo Pareto stumbled upon something no one had seen before.

See, he suspected there might be a hidden pattern in how much money people make.

So he gathered income tax records from Italy, England, France, and other European countries.

And for each country, he plotted the distribution of income.

Each country he looked at, he saw the same pattern.

A pattern which still holds in most countries to this day.

And it's not a normal distribution.

If you think about a normal distribution like height, there's a clearly defined average.

And extreme outliers basically never happen.

I mean, you're never going to find someone who is, say, five times the average height.

That would be physically impossible.

But Pareto's income distributions were different.

Take this curve for England.

It shows the number of people who earn more than a certain income.

The curve starts off declining steeply.

Most people earn relatively little.

But then it falls away gradually, much more slowly than a normal distribution would.

And it spans several orders of magnitude.

There were people who earned 5 times, 10 times, even 100 times more than others.

That kind of spread just wouldn't happen if income were normally distributed.

Now, to shrink this huge spread of data, Pareto calculated the logarithms of all the values and plotted those instead.

In other words, he used a log -log plot.

And when he did that, the broad curve transformed into a straight line.

The gradient was around negative 1 .5.

That means each time you double the income, say from 200 pounds to 400 pounds,

the number of people earning at least that amount drops off by a factor of 2 to the power of 1 .5, which is around 2 .8.

And this pattern holds for every doubling of income.

So Pareto could describe the distribution of incomes with one simple equation.

The number of people who earn an income greater than

or equal to x is proportional to 1 over x to the power of 1 .5.

Now that's what Pareto saw for England, but he performed the same analysis on data from Italy, France, Prussia, and a bunch of other countries.

And he saw the same thing again and again.

Each time the data transformed into a straight line.

And the gradients were remarkably similar.

That meant Pareto could describe the income distribution in each country with the same equation, one over the income to some power,

where that power is just the absolute gradient of the logarithmic graph.

This type of relationship is called a power law.

When you move from the world of normal distributions to the world of power laws, things change dramatically.

So to illustrate this, let's take a trip to the casino to play three different games.

At table number one, you get 100 tosses of a coin.

Each time you flip and it lands on heads, you win $1.

So the question is, how much would you be prepared to pay to play this game?

Well, we need to work out how much you'd expect to win in this game

and then pay less than that expected value.

So the probability of throwing a head is one half.

Multiply that by $1 and multiply that by 100 tosses, that gives you an expected payout of $50.

So you should be willing to pay anything less than $50 to play this game.

Sure, you might not win every time, but if you play the game hundreds of times, the small variations either side of the average will cancel out and you can expect to turn a profit.

One of the first people to study this kind of problem was Abraham de Moivre in the early 1700s.

He showed that if you plot the probability of each outcome, you get a bell -shaped curve, curve, which was later coined the normal distribution.

Normal distributions, the traditional explanation is that when there are a lot of effects that that are random.

that are adding up, that's when you expect normals.

So like how tall I am depends on a lot of random things about my nutrition, about my parents' genetics, all kinds of things,

but if these random effects are additive, that is what tends to lead to normals.

At table number two, there's a slightly different game.

You still get 100 tosses of the coin, but this time, instead of potentially winning a dollar on each flip, your winnings are multiplied by some factor.

So you start out with $1, and then every time you toss ahead, you multiply your winnings by 1 .1.

If instead the coin lands on tails, you multiply your winnings by 0 .9.

And after 100 tosses, you take home the total.

That is, the dollar you started with times the string of 1 .1s and 0 .9s.

So how much should you pay to play this game?

Well, on each flip, your payout can either grow or shrink

and each is equally likely each time you toss the coin

so the expected factor each turn is just 1 .1 plus 0 .9 divided by 2

which is 1 so

if you start out with one dollar then your expected payout is just one dollar

that means you should be willing to pay anything less than a dollar to play this game right well

if you look at the distribution of payouts you can see that you could win big If you tossed 100 heads, you'd win 1 .1 to the power of 100.

That's almost $14 ,000.

Although the chance of that happening is around 1 in 10 to the power of 30.

You'd be more likely to win the lottery three times in a row.

On the other hand, the median payout is around $0 .61.

So if you're only playing the game one time and you want even odds of turning a profit, well then you should pay less than $0 .61.

Though either way, if you played the game hundreds of times, your payout would average out to one dollar.

Now watch what happens if we switch the x -axis from a linear scale to a logarithmic scale.

Well then you see the curve transforms into a normal distribution.

That's why this type of distribution is called a log normal distribution.

When random effects multiply, if I have a certain wealth

and then my wealth goes up by a certain percentage next year because of my investments,

and then the year after that it changes by another random factor As opposed to adding, I'm multiplying year after year.

If you have a big product of random numbers, when you take the log of a product, that's the sum of the logs.

So what was a product of random numbers then gets translated into sums of logs of random numbers.

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And that's what leads to this so -called log -normal distribution.

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And log -normal distributions produce big inequalities.

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You don't just see a mean, you see a mean with a big long tail it's much

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more likelihood of really big events in this case tremendous wealth being obtained than you would expect from a normal distribution.

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The reason this curve is so asymmetric is because the downside is capped at zero, so at most you could lose one dollar.

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But the upside can keep growing up to nearly $14 ,000.

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Now let's go on to table three.

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Again you'll be tossing a coin, but this time you start out with a dollar

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and the payout doubles each time you toss the coin

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and you keep tossing until you get a heads then the game ends

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so if you get heads on your first toss you get two dollars

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if you get a tails first

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and then hit a heads on your second toss you get four dollars

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if you flipped two tails

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and then a head on your third toss you'd get eight dollars

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and so on

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if it took you to the nth toss to get a heads you would get two to the n dollars

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so how much should you pay to play this game well

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as in our previous example we need to work out the expected value

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so suppose you throw a head on your first try the payout is two dollars

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and the probability of that outcome is a half

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so the expected value of that toss is a dollar

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if it takes you two tosses to get a heads then the payout is four dollars

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and the probability of that happening is one over four

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so again the expected value is one dollar we also need to add in the chance

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that you flip heads on your third try in

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that case the payout is eight dollars and the probability of that happening is 1 over 8.

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So again, the expected value is $1.

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And we have to keep repeating this calculation over all possible outcomes.

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We have to keep adding $1 for each of the different options for flipping the coin, say, 10 times until it lands on heads or 100 times before you get heads.

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I know it's extremely unlikely, but the payout is so huge that the expected value of that outcome is still $1.

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So it still increases the expected value of the whole game this means

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that theoretically the total expected value of this game is infinite this is known as the saint petersburg paradox

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if you look at the distribution of payouts you can see it's uncapped.

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It spans across all orders of magnitude.

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You could get a payout of $1 ,000, $100 ,000, or even $1 ,000 ,000 or more.

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And while a million -dollar payout is unlikely, it's not that unlikely.

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It's around one in a million.

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Now, if you transform both axes to a log scale, you see a straight line with a gradient of negative one.

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The payout of the St. Petersburg paradox follows a power law.

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The specific power law in this case is

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that the probability of a payout x is equal to x to the power of negative 1, or 1 over x.

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In the previous games, when you have a normal distribution, or even a log normal distribution, you can measure the width of that distribution, its standard deviation.

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And in a normal distribution, 95 % of the data fall within two standard deviations from the mean.

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But with a power law, like in the St. Petersburg paradox, there is no measurable width.

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The standard deviation is infinite.

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This makes power laws a fundamentally different beast with some very weird properties.

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Imagine you take a bunch of random samples and then average them and then take more random samples and average them.

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You'll find that the average keeps going up.

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It doesn't converge.

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The more you measure, the bigger the average is, which is really weird.

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It sounds impossible but it's

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because it has such a heavy tail meaning the probability of really whopping big events is

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so significant that if you keep measuring occasionally you're going to measure one of those extreme outliers

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and they will totally skew the average it's sort of like saying you know

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if you're standing in a room with bill gates

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or elon musk the average wealth in that room is going to be $100 billion or something

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because the average is dominated by one outlier.

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And that same idea, one outlier can dominate the average, shows up online too.

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A handful of companies, servers, and data centers hold the personal information of millions of people.

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So when one of them gets hacked, it can have ripple effects across the whole network.

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We've had scammers get a hold of email addresses

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and phone numbers of writers on our team and then send them messages pretending to be me.

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And now back to PowerLaws.

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So why do you get a PowerLaw from the simple St. Petersburg setup?

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If you look at the payout, x, you can see it grows exponentially with each toss of the coin, x equals 2 to the n.

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But if you look at the probability of tossing the coin that many times to get a heads, you can see that this probability shrinks exponentially.

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So the probability of flipping a coin n times is a half to the power of n.

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But we're not really interested in the number of tosses, we're interested in the payout.

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Now, we know that x equals 2 to the n, so instead of writing 2 to the n in our probability equation, we can just write x.

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So we end up with this.

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The probability of a payout of x dollars is equal to 1 over x, or in other words, x to the power of negative 1.

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You put them together, the exponentials conspire to make a power law.

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And that's a very common thing in nature, that a lot of times when we see power laws, there are two underlying exponentials that are dancing together to make a power run.

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One example of this is earthquakes.

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If you look at data on earthquakes, you find that small earthquakes are very common.

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But earthquakes of increasing magnitudes become exponentially rarer.

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But the destruction that earthquakes cause is not proportional to their magnitude.

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It's proportional to the energy they release.

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And as earthquakes grow in magnitude, that energy grows exponentially.

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So there's this exponential decay in frequency of earthquakes of a given magnitude

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and an exponential increase in the amount of energy released by earthquakes of a certain magnitude.

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So when you combine those two exponentials to eliminate the magnitude, what you find is a power law.

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But power laws also reveal something deeper about the underlying structure of a system.

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To see this in action, let's go back to the third coin game and the St. Petersburg paradox.

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Now, you can draw all the different outcomes as a tree diagram, where the length of each branch is equal to its probability.

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So starting with a single line of length one, and then a half for the first - two branches, a quarter for the next four, and so on.

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Now, when you zoom in, you keep seeing the same structure repeating at smaller and smaller scales.

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It's self -similar, like a fractal.

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And that's no coincidence.

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We see the same fractal -like pattern in the veins on a leaf, river networks, the blood vessels in our lungs, even lightning.

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And in all of these cases, we can describe the pattern with a power law.

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Power laws and fractals are intrinsically linked.

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That's because power laws reveal something fundamental about a system's structure.

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So I've got a magnet and I've got a screw.

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And you'll notice if I bring them close together, then the screw gets attracted to the magnet.

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And that's because there's a lot of iron in it, which is ferromagnetic.

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But watch what happens if I start heating this up.

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Trying to, oh, you see that?

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Oh, there it went.

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There it went.

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You see, you heat it up and and suddenly it becomes non -magnetic.

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To find out what happened, let's zoom in on this magnet.

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Inside a magnet, each atom has its own magnetic moment, which means you can think of it like its own little magnet or compass.

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If one atom's moment points up, its neighbors tend to point that way too, since this lowers the system's overall potential energy.

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Therefore, at low temperatures, you get large regions called domains where all the moments align.

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And when many of these domains also align, their individual magnetic fields reinforce to create an overall field around the magnet

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but if you heat up the magnet each atom starts vibrating vigorously the moments flip up

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and down and so the alignment can break down and

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when all the moments cancel out then there's no longer a net magnetic field now

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if you have the right equipment you can balance any magnetic material right on

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that transition point right between magnetic and non -magnetic this is called the critical point

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and it occurs at a specific temperature called the Curie temperature.

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I asked Casper and the team to build a simulation to show what's going on inside the magnet at this critical point.

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Each pixel represents the magnetic moment of an individual atom.

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Let's say red is up and blue is down.

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Now, when the temperature is low, we get these big domains where the magnetic moments are all aligned and you get an overall magnetic field.

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But if we really crank up the temperature, then all of these moments start flipping up and down and

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so they cancel out and the magnet loses its magnetism

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so that's exactly what happened in our demo but

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if we tune the temperature just right right to

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that curie temperature then the pattern becomes way more interesting this

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looks like a map like a map yeah it almost looks like the mediterranean

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or something it's almost stable like atoms that are pointing one way tend to point that way for a while.

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But there is clearly fluctuations as well, so domains are constantly coming and going.

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It's both got some elements of stability and some persistence over time, some features which are consistent,

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but it's also not locked in place because you notice changes over time.

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If you zoom in, you find that the same kinds of patterns repeat at all scales.

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You've got domains of tens of atoms, hundreds, thousands, even millions.

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There's just no inherent scale to the system.

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That is, it's scale -free.

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It's just like a fractal.

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And if you plot the size distribution of the domains, you get a power law.

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The underlying geometry suddenly shows a fractal character that it doesn't have on either side of the phase transition.

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Right at the phase transition, you get fractal behavior and that pops out as a power law." In fact, whenever you find a power law,

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that indicates you're dealing with a system that has no intrinsic scale.

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And that is a signature of a system in a critical state, which it turns out has huge consequences.

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See, normally in a magnet, below the Curie temperature, each atom influences only its neighbors.

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If one atom's magnetic moment flips up, then that means that its neighbors are slightly more likely to point up two.

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But that influence is local.

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It dies out just a few atoms away.

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But as the magnet approaches its critical temperature those local influences start to chain together one spin nudges its neighbor

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and that neighbor nudges the next and

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so on like a rumor spreading through a crowd

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and the result is that the effective range of influence keeps expanding

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and right at the critical point it becomes effectively infinite a flip on one side can cascade throughout the entire material

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so you get these small causes just a single flip to reverberate throughout the entire system

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and it gets right into that

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that point where the system is maximally unstable anything can happen

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it's also maximally interesting in a way it's it means the system is most unpredictable

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most uncertain it's really hard to know what's going to happen next

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and that seems to be a natural procedure that happens in many different systems one such system is forest fires

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In June 1988, a lightning strike started a small fire near Yellowstone National Park.

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This was nothing out of the ordinary.

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Each year, Yellowstone experiences thousands of lightning strikes.

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Most don't cause fires, and those that do tend to burn a few trees, maybe even a few acres, before they fizzle out.

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Three quarters of fires burn less than a quarter of an acre.

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The largest fire in the park's recent history occurred in 1931.

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That burned through 18 ,000 acres, an area slightly larger than Manhattan.

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But the 1988 fire was different.

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That initial spark spread, slowly at first covering several thousand acres.

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Then, over the next couple of months, it merged with other small fires to create an enormous complex of megafires that blazed across 1 .4 million acres of land.

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That's around the size of the entire state of Delaware.

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That's 70 times bigger than the previous record, and 50 times the area of all the fires over the previous 15 years combined.

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So what was so special about the 1988 fires?

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Well, to find out, we made a forest fire simulator.

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We've got a grid of squares.

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And on each square, either a tree could be there, it could grow, or it could not be there.

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There's going to be some probability for lightning strikes.

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So, you know, the higher the probability, the more fires we're going to have we can run this

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so trees are growing trees are growing forest is filling in

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nice getting pretty dense what do you expect is gonna happen

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I expect to see some fires probably you know now

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that oh that was good

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that was a good little fire whoa whoa no way oh

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that's crazy you haven't adjusted the parameters right it's just like

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not yet not yet this seems like a very critical situation just by itself i say

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that because of how big

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that fire was this sort of system will tune itself to criticality

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and you can you can see it start to happen

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so right now i think is a good moment where you have basically domains of a lot of different sizes

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and then one way to think about it is

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if some of these domains become too big then you get a single fire like

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that one perfectly timed burns it's gonna propagate throughout the whole thing

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and burn it back down a little but then

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if it goes too hard then now you've got all these domains where there are no trees

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and so it's gonna you know grow again to bring it back to

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that critical state i can see how it's the feedback mechanism right

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that the fire gets rid of all the trees

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and there's nothing left to burn and then

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that has to fill in again yeah yeah but

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if there hasn't been a fire then the forest gets too thick and then it's ripe for this sort of massive

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fire for a magnet you have to painstakingly tune it to the critical point

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but the forest naturally drives itself there this phenomenon is called self -organized criticality yeah

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and if you let it run what you get is again a power law distribution

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so this is log log so it should be a straight line

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that kind of stuff seems so totally random and unpredictable

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and it is in one way

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and yet it follows a pattern there's a consistent mathematical pattern

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to all these kind of disasters um it's it's shocking is

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there something fractal about this mostly in terms of the I guess domains of the trees

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when you're at that critical state

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so you get very dense areas you get non -dense areas

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and as a result

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when a single lightning bolt strikes you can get fires of

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all sizes most often you get small fires of 10

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or fewer trees burning a little less frequently you get fires of less than 100 trees

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and then every once in a while you get these massive fires

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that reverberate throughout the entire system now you might expect that

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because the fire is so large there has to be a significant event causing it.

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But that's not the case.

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Because the cause for each fire is the exact same.

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It's a single lightning strike.

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The only difference is where it strikes and the exact makeup of the forest at that time.

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So, in some very real way, the large fires are nothing more than magnified versions of the small ones.

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And even worse, they're inevitable.

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So, what we've learned is that for systems in a critical state, there are no special events causing the massive fires.

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There was nothing special about the Yellowstone fire.

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In 1935, the U .S.

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Forest Service established the so -called 10 AM policy.

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The plan was to suppress every single fire by 10 AM on the day following its initial report.

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Now, naively, this strategy makes sense.

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I mean, if you keep all fires under strict control, then none can ever get out of hand.

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But it turns out this strategy is extremely risky.

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So let's say we're going to bring down the lightning probability, so it's very small, only one in a million right now.

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And we're also going to crank up, you know, the tree growth a little bit.

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Now, what do you think is going to happen?

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We're going to get some big fires, I would imagine.

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Like a lot of not fire and then some huge fires.

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Yeah.

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Yep.

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Oh, boy.

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So nowadays the fire service has a very different approach.

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They acknowledge that some fires are essential to make the megafires less likely.

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So, they let most small fires burn and only intervene when necessary.

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In some cases, they even intentionally create small fires to burn through some of the buildup.

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Though it could take years to return the forest to its natural state after a century of fire suppression.

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But it's more than just the Earth's forests that are balanced in this critical state.

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Every day, the Earth's crust is moving and rearranging itself.

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Stresses build up slowly as tectonic plates rub against each other.

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Most of the time, you get a few rocks crumbling.

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The ground might move just a fraction of a millimeter.

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But the stresses dissipate in many earthquakes that you wouldn't even feel.

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There are really tiny earthquakes that are happening right now beneath your feet.

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You just can't feel them because they're very small.

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But they are earthquakes.

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They're driven by small, slipping movements in the Earth's crust.

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But sometimes those random movements can trigger a powerful chain reaction.

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In Kobe, Japan, the morning of January 17th, 1995 seemed just like any other.

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This was a peaceful city, and although Japan as a country is no stranger to earthquakes, Kobe hadn't suffered a major quake for centuries.

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Generations grew up believing the ground beneath them was stable.

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But that morning, deep underground, a stress released nearby the Nojima fault line.

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The stress propagated to the next section of the fault and the next.

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Within seconds, the rupture cascaded along 40 kilometers of crust, shifting the ground by up to two meters and releasing the energy equivalent of numerous atomic bombs.

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The resulting quake destroyed thousands of homes, along with most major roads and railways leading into the city.

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It killed over 6 ,000 people and forced 300 ,000 from their homes.

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How far it goes depends a lot on chance and the organization of all that stress field in the Earth's crust.

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And it just seems to be organized in such a way

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that it is possible oftentimes for the earthquake to trickle along an avalanche along a long way

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and produce a very large unusual earthquake.

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But if you look at the process behind that earthquake, it is exactly the same physical process.

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It's just that the earthquake generating process naturally produces events that range over an enormous range of scale, and we're not really used to thinking about that.

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We have this ingrained assumption that we can use the past to predict the future.

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But when it comes to earthquakes or any system that's in a critical state, that assumption can be catastrophic because they're famously unpredictable.

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So how can you even begin to model something like the behavior of earthquakes?

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In 1987, Danish physicist Per Back and his colleagues considered a simple thought experiment.

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Take a grain of sand and drop it on a grid, then keep dropping grains on top until at some point the sand pile gets

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so steep that the grains tumble down onto different squares.

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What they looked at was the size of these, what they were calling avalanches, these reorganizations of numbers of grains of sand.

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They asked for how often do you see avalanches of a certain size?

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This is the most simple version of a sand pile simulator

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that you could almost imagine we're gonna drop a little grain of sand at first always in the center

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and then it's just gonna keep going up for one grain

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it'll be fine for two grains it'll be fine three grains it'll be fine

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but it's on the edge of toppling and then when it reaches four or more it's gonna basically go

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feels a bit like a i don't know pulsing thing like something's trying to escape

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or something very video game like that seemed pretty crazy and it is symmetrical yeah nice geometric features

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So this might be interesting, because right now we've passed it at a point where this middle one is going to go.

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And then you look around it and you see,

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essentially you can think of these brown or these three tall grain stacks as being maximally unstable.

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They're about to go.

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And so you can think of them as these fingers of instability.

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If anything touches them, the whole system, like they're just going to go.

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I see it propagating out.

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It's cool seeing it slower.

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I feel like you can see several waves propagating at the same time.

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Some people have reasoned that the Earth's crust becomes riddled with similar fingers of instability, where you get stresses building up.

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And then when one rock crumbles, it can propagate along these fingers, potentially triggering massive earthquakes.

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If you look at the data, there's some.. even more compelling evidence that links the sand pile simulation to earthquakes.

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Let's say instead of dropping it at the center, pretty unrealistic to have it drop in the center,

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I'm going to drop at random. That is crazy.

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You can actually see it tune itself to the critical state.

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Like at the start, you only see these super tiny avalanches.

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And then now it's everything.

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There are events of all sizes.

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One grain of sand might knock over just a few others, or it could trigger an avalanche of millions of grains that cascade throughout the entire system.

407

And if you look at the power law you get from the sandpal simulation, it closely resembles the power law of the energy released by real earthquakes earthquakes.

408

But if you look at the sandpile experiment more closely, it doesn't just resemble earthquakes.

409

What does it remind you of?

410

Forest fires.

411

Right?

412

Feels like it's the exact same behavior.

413

That's the really surprising thing.

414

And that's why this little paper with a sandpile was published in the world's top journal.

415

Because it did something that people just didn't really think was possible.

416

Now what's ironic is, if you look at real sandpiles, they don't behave like this.

417

okay you said sand i'm gonna do an experiment on a real sand pile

418

and of course it doesn't follow a power law distribution of avalanches at all it's totally wrong

419

pair buck naturally gets a chance to reply to the criticism

420

and he says i'm pretty close to quoting he says self -organized criticality only applies to the systems it applies to

421

so he doesn't care the fact

422

that his theory is not relevant to real sand piles

423

so what get out of my face he's interested in bigger

424

fish to fry than than you know sand piles it's like

425

you're taking me too literally i'm talking about a universal mechanism for generating power laws

426

and the fact

427

that it doesn't depend it doesn't work in real sand is uninteresting to him I thought that took some real nerve.

428

You could think about the Earth and the Earth going around the Sun.

429

That's a very complex system.

430

You've got the molten core, everything slashing around, then you've got oceans, and you've even got the Moon going around the Earth, which in theory, you know,

431

all should affect the exact motion of the Earth around the Sun.

432

But Newton ignored all of

433

that all he looked at was just a single parameter essentially the mass of the earth and with

434

that he could correctly for the most part predict how the

435

earth was going to go around the sun similarly here there are people

436

that have looked at these phenomena

437

that go to the critical state in this in this case it's self -organized criticality is it brings itself there

438

and what they find is

439

that there's this universal behavior where it doesn't even really matter

440

what the sub parts are you You just get the behavior that's the exact same.

441

At that critical point, when all the forces are poised and the system is right on that delicate balance between being organized,

442

highly organized, or being totally disorganized, it turns out that almost none of the physical details about that system matter to how it behaves.

443

There's just a universal behavior that is irrespective of what physical system you're talking about.

444

The term that was used is called universality.

445

And it's kind of a miracle.

446

It means you can make extremely powerful theories without involving any technical details, any real details of the material.

447

What this means is that you could have these systems that on the surface seem totally different.

448

But when you get to the critical point, they all behave in the exact same way.

449

The other thing you could do is instead of this being trees, you could imagine it being people.

450

And the thing that's spreading is disease.

451

Is disease, yeah.

452

You almost get something for nothing at these critical points.

453

See, many of these systems fall into what's known as universality classes.

454

Some of them you need to tune to get there, like magnets at their curie temperature, or fluids like water or carbon dioxide at their critical point.

455

But some other systems seem to organize themselves to criticality, like the forest fires or sand piles or earthquakes.

456

But what's crazy is that if you succeed in understanding just one system from a class, then you know how all the systems in that class behave.

457

And that includes even the crudest, simplest toy models, like the simulations we've looked at.

458

So you can model incredibly complex systems with the most basic of models.

459

And some people think this critical thinking applies even further.

460

When we look around the world, there are lots of systems that show the same power law behavior that we see in this.

461

Thank you critical systems.

462

It's in everything from DNA sequencing to the distribution of species in an ecosystem to the size of mass extinctions throughout history.

463

We even see the same behavior in human systems, like the populations of cities, fluctuations in stock prices, citations of scientific papers,

464

and even the number of deaths in wars.

465

So some people argue that these systems, and perhaps many parts of our world, also organize themselves to this critical point.

466

So the fact that all these natural hazards, as they call them, floods, wildfires, and earthquakes, they all follow power law distributions means that these extreme events

467

are much more common than you would think based on normal distribution thinking.

468

If you find yourself in a situation or an environment that is sort of governed by a power law, how should you change your behavior?

469

If you have events with one of these power distributions what you're seeing most of the time is small events

470

and this can lull you into a full sense of security

471

you think you understand how things are going you know floods for example there are a lot of small floods

472

and then every once in a

473

while there's a huge one response to this is insurance

474

that insurance is designed precisely to protect you against the large rare events

475

that would otherwise be very bad but then there's the other side of that picture which is

476

You're the insurance company that needs to insure people

477

and they have a particularly difficult job

478

because they have to be able to say how much to charge

479

so that they have enough money to pay out when the big bad thing comes along.

480

In 2018, a forest fire tore through Paradise, California.

481

It became the deadliest and most destructive fire in the state's history.

482

But the insurance company, Merced Property and Casualty, hadn't planned for something that huge.

483

And when the claims came in, they just didn't have the reserves to pay out.

484

So just like that, the company went bust.

485

But while extreme events can cripple some companies, there are entire industries that are built on parallel distributions.

486

Between 1985 and 2014, private equity firm Horsley Bridge invested in 7 ,000 different startups,

487

and over half of their investments actually lost money.

488

But the top 6 % more than 10x in value and generated 60 % of the firm's overall profit.

489

In fact, the best venture capital firms often have more investments that lose money.

490

They just have a few crazy outliers that show extraordinary growth, a few outliers that carry the entire performance. In 2012,

491

Y Combinator calculated that 75 % of their returns came from just two out of the 280 startups they invested in.

492

So, venture capital is a world that depends on taking risks in the hope

493

that you'll get a few of these extreme outliers which outperform all of the rest of the investments combined.

494

Book publishers operate in a similar fashion.

495

Most titles flop, but in 1997, a small independent UK publisher called Bloomsbury took a chance on a story about a boy wizard.

496

The boy's name, of course, was Harry Potter.

497

And now Bloomsbury is a globally recognized brand.

498

We see a similar pattern play out on streaming platforms.

499

On Netflix, the top 6 % of shows account for over half of all viewing hours on the platform.

500

On YouTube, less than 4 % of videos ever reach 10 ,000 views, but those videos account for over 93 % of all views.

501

All these domains follow the same principle that Pareto identified over 100 years ago, where the majority of the wealth goes to the richest few.

502

The entire game is defined by the rare runaway hits.

503

But not every industry can play this game.

504

Like if you're running a restaurant, you need to fill tables night after night.

505

You can't have one particularly busy summer evening

506

that brings in millions of customers to make up for a bunch of quiet nights.

507

Over a year, the busy nights and quiet ones balance out, and you're left with the average.

508

Airlines are similar.

509

An airline needs to fill seats on each flight.

510

You can't squeeze a million passengers onto one plane.

511

So it's the average number of passengers over the year that defines an airline's success.

512

We're used to living in this world of normal distributions and you act a certain way, but as soon as you switch to this realm that is governed by a power law,

513

you need to start acting vastly different.

514

It really pays to know what kind of world you're, or what kind of game you're playing.

515

That is good, that's good, yes.

516

You should come on camera and just say that, just like that.

517

If you're in a world where random additive variations cancel out over time, then you get a normal distribution.

518

And in this case, it's the average performance, so consistency, which is important.

519

But if you're in a world that's governed by a power law where your returns can multiply

520

and they can grow over many orders of magnitude, then it might make sense to take some riskier bets in the hope that one of them pays off huge.

521

In other words, it becomes more important to be persistent than consistent.

522

Though, as we saw in the second coin game, totally random multiplicative returns give you a log normal distribution, not a power law.

523

To get a power law, there must be some other mechanism at play.

524

In the early 2000s, Albert Laszlo Barbasci was studying the internet.

525

And to his surprise, he found that there was no normal web page with some average number of links.

526

Instead, the distribution followed a power law.

527

A few sites, like Yahoo, had thousands of times more connections than most of the others.

528

Birabashi wondered what could be causing this power law of the internet, so he made a simple prediction.

529

As new sites were added to the internet, they were more likely to link to well -known pages.

530

To test this prediction, he and his colleague, Reika Albert, ran a simulation.

531

They started with a network of just a few nodes, and gradually they added new nodes to the network, with each new node more likely to connect to those with the most links.

532

As the network grew, a power law emerged.

533

The power was around negative two, which almost exactly matched the real data of the internet.

534

Look at that.

535

That's fun.

536

It's still so satisfying.

537

This will basically also distribute a power law.

538

One of the ideas here is that, you know, this could be individuals or even companies.

539

And so if you're more likely to become more successful or more well -known, the more known or successful you already are,

540

you're going to get this sort of runaway effect where you you get a few

541

that sort of dominate you know the distributions i wonder

542

if part of the takeaway is like if you're playing some sort of game

543

that is dominated by power law then you better do the work as much of it as early as possible

544

so you get to benefit from the snowball effect essentially yeah

545

i guess i guess that's that's a good idea i'm not

546

sure whether you can control it though human beings like to think of ourselves as being a bit special

547

and that maybe somehow because we're intelligent and have free will we will escape the

548

provenance of the laws of physics in order in organization

549

but I think that's probably not not the case so

550

if you look at at the number of world wars and

551

if you make a crude measure of how big is the world war by how many people it kills

552

which is a bit macabre but still you find that again it follows a power law.

553

Virtually identical to the power law you find in stock market crashes." So if the world is shaped by power laws,

554

then it feels like we're poised in this kind of critical state, where two identical grains of sand, two identical actions can have wildly different effects.

555

Most things barely move the needle, but a few rare events totally dwarf the rest, and that I think is the most important lesson.

556

If you choose to pursue areas governed by the normal distribution, you can pretty much guarantee average results.

557

But if you select pursuits ruled by power laws, the goal isn't to avoid risk, it's to make repeated intelligent bets.

558

Most of them will fail, but you only need one wild success to pay for all the rest.

559

And the thing is that beforehand, you cannot know which bet it's going to be, because the system is maximally unpredictable.

560

It could be that your next bet does nothing, it could do a little bit, or it could change your entire life.

561

In fact, around three years ago, I was reading this little book.

562

And in the book, there was this little line saying something like, one idea could transform your entire life.

563

So right underneath that, I wrote, send an email to Veritasio.

564

A couple days later, I wrote an email to Derek saying, hey, Derek, I'm Casper, I study physics, and I can help you research videos.

565

I didn't hear back for four weeks, so I was getting pretty sad and just wanted to forget about it

566

and move on

567

but then a couple days later i got an email back

568

saying hey casper we can't do an internship right now

569

but how would you like to research write

570

and produce a video as a freelancer so i did and that's how i got started at paratassia

571

hey just a few quick final things all the simulations

572

that we used in this video So we'll make available for free for you to use in the link in the description.

573

And the other thing is that we just launched the official Veritasium game.

574

It's called Elements of Truth, and it's a tabletop game with over 800 questions.

575

It's the perfect way to challenge your friends and see who comes out on top.

576

Now at Veritasium, we're all quite competitive, so every time we play, things get a little bit heated, but that's honestly a big part of the fun.

577

Now, when we launched on Kickstarter, we got a lot of questions asking if we could ship to specific countries.

578

And originally we didn't enable this, and this is our mistake, this is on us, and we totally hear you.

579

But I'm glad to say that right now we have enabled worldwide shipping.

580

So no matter where you are in the world, you can get your very own copy.

581

To reserve your copy and get involved, scan this QR code or click the link in the description.

582

I want to thank you for all your support, and most of all, thank you for watching.

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人生のゲームの「正しい」遊び方？ビデオが教える意外な法則

このビデオでは、身の回りの現象が「正規分布」と「べき則」の2つの異なるパターンで動いていることが説明されています。身長やIQは平均値の周りに集中する正規分布ですが、所得や戦争の死者数などはべき則に従い、極端な値が出現しやすいのです。パレートが発見した所得分布の法則は、「所得がX倍になると、それ以上稼いでいる人数は1/X^1.5に比例する」というもので、これはべき則の典型例です。カジノのゲームを例に挙げると、コイントスのような正規分布のゲームは平均値が安定していますが、べき則のゲームは予測が難しく、大きな変動が起こります。

実用的なフレーズとコロケーション

normal distribution：正規分布。「身長はnormal distributionに従う」のように使います。
power law：べき則。「所得分布はpower lawに従う」と表現します。
expected value：期待値。「このゲームのexpected valueは50ドルです」と言います。
log-log plot：対数-対数プロット。「log-log plotでグラフを描くと直線になる」と説明します。
outliers：外れ値。「正規分布ではoutliersはほとんどありません」と使います。

あなたのシャドーイングチャレンジ

英語シャドーイングやshadowspeakは、スピーキング力を上げるのに最も効果的な方法の一つです。さっそく動画で英語学習を始めましょう！ビデオの中で「The system you're looking at doesn't have any inherent physical scale. It's really hard to know what's gonna happen next.」という部分があります。この2文を選び、以下のステップで練習してみてください。
1. ビデオを再生し、1文ずつ聞き取ります。
2. すぐに同じ速さ、同じイントネーションで繰り返します（shadow speak）。
3. 5回繰り返したら、自分で録音して元の音声と比較しましょう。
この練習を続けると、英語スピーキング練習が楽しくなり、自然な発音とリズムが身につきますよ！

この動画の文法

話し手がよく使っている文型を、動画の実際の表現とともに紹介します。

文型	動画での表現
受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる	is called · are multiplied · being obtained
現在完了形 have/has + 過去分詞 — 過去の出来事が今も関係している	We've had · haven't adjusted · hasn't been
関係詞節 who / which + 節 — 人や物について情報を加える	someone who is · people who earn · people who earned

シャドーイングとは？英語上達に効果的な理由

シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。

シャドーイングのやり方: ステップ別の完全ガイドを読む →

シャドーイング練習: You've (Likely) Been Playing The Game Of Life Wrong - 動画で英語スピーキングを学ぶ

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人生のゲームの「正しい」遊び方？ビデオが教える意外な法則

実用的なフレーズとコロケーション

あなたのシャドーイングチャレンジ

この動画の文法

シャドーイングとは？英語上達に効果的な理由