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蝴蝶效应背后的科学

The Science Behind the Butterfly Effect

Part of this video is sponsored by LastPass.
本视频由LastPass部分赞助播出
More about last pass at the end of the show.
更多有关LastPass的信息请见视频结尾
The butter fly effect is the idea that the tiny causes,
蝴蝶效应是指很小的事件可能会带来巨大的影响
like a flap of a butter fly’s wings in Brazil,
比如 巴西的一只蝴蝶扇动了翅膀
can have huge effects, like setting off a tornado in Texas
可能会在德克萨斯州引发一场龙卷风
Now that idea comes straight from the title of a scientific paper
这个理论来自于近50年前发表的
published nearly 50 years ago
一篇科学论文的标题
and perhaps more than any other recent scientific concept,
也许比其他任何近期的科学概念
it has captured the public imagination
更吸引大众的想象力
I mean on IMDB there is not one but 61different movies,
在IMDB上 有61部不同的电影
TV episodes, and short films with ‘butterfly effect’ in the title
电视剧和短片的标题带有“蝴蝶效应”
not to mention prominent references in movies like Jurassic Park,
更不用说像《侏罗纪公园》是歌里 书里 还有表情包
or in songs, books, and memes.
电影里有显著的引用
Oh the memes
哦 大多数表情包
in pop culture the butterfly effect has come to mean
在流行文化中 蝴蝶效应已经意味着
that even tiny, seemingly insignificant choices you make
即使是微小的 看起来微不足道的选择
can have huge consequences later on in your life
也会对你以后的生活造成巨大影响
and I think the reason people are so fascinated by the butterfly effect is
我认为人们之所以对蝴蝶效应这么着迷
because it gets at a fundamental question
是因为它触及到了一个基本问题:
Which is, how well can we predict the future?
我们怎样准确地预测未来?
Now the goal of this video is to answer that question
本视频的目的是通过研究
by examining the science behind the butterfly effect
蝴蝶效应背后的科学来回答这个问题
so if you go back to the late 1600s,
如果你回到17世纪末
after Isaac Newton had come up with his
在艾萨克·牛顿提出了
laws of motion and universal gravitation,
运动定律和万有引力后
everything seemed predictable.
万物似乎皆可预测
I mean we could explain the motions of all the planets and moons,
我的意思是我们能解释所有行星和卫星的运动
we could predict eclipses and the appearances of comets
并提前数个世纪精确地预测出
with pinpoint accuracy centuries in advance
月食和彗星的出现
French physicist Pierre-Simon Laplace
法国物理学家皮埃尔·西蒙·拉普拉斯
summed it up in a famous thought experiment:
提出了一个非常著名的思想实验:
he imagined a super-intelligent being, now called Laplace’s demon,
他假设有一个现在被称为拉普拉斯妖的智者
that knew everything about the current state of the universe:
知道宇宙现有状态的所有事情:
the positions and momenta of all the particles
所有粒子的位置和动量
and how they interact
以及它们如何反应
if this intellect were vast enough to
他总结道 如果这个智者
submit the data to analysis, he concluded,
能够十分广泛地分析这些数据
then the future, just like the past, would be present before its eyes
那么未来就像过去一样 呈现在它眼前
This is total determinism:
这就是完全决定论:
the view that the future is already fixed,
一种认为未来已经固定的观点
We just have to wait for it to manifest itself
我们只需要等待它发生
I think if you’ve studied a bit of physics,
我想如果你学过一点物理
this is the natural viewpoint to come away with
就知道这偏离了自然观
I mean sure there’s Heisenberg’s uncertainty principle
我的意思是 在量子力学中确实有
from quantum mechanics,
海森堡的不确定性原理
but that’s on the scale of atoms;
但那是在原子层面上
Pretty insignificant on the scale of people.
在人的层面上相当微不足道
Virtually all the problems I studied were ones
几乎所有我研究的问题
that could be solved analytically
都可以通过分析解决
like the motion of planets, or falling objects, or pendulums
比如行星的运动 或是下落的物体 或是钟摆
and speaking of pendulums
说到钟摆
I want to look at a case of a simple pendulum here
我想在这里看一个简单的钟摆案例
to introduce an important representation of dynamical systems,
来介绍动态系统的重要表现形式
which is phase space
即相空间
so some people may be familiar with position-time or velocity-time graphs
有些人可能对位置时间图或是速度时间图比较熟悉
but what if we wanted to make a 2d plot
那如果我们想做一个
that represents every possible state of the pendulum?
展现钟摆所有可能状态的2D图呢?
Every possible thing it could do in one graph
在一个图形中表示每一件可能的事情
well on the x-axis we can plot the angle of the pendulum,
我们可以在x轴上绘制钟摆的角度
and on the y-axis its velocity.
在y轴上绘制钟摆的速度
And this is what’s called phase space.
这就是所谓的相空间
If the pendulum has friction it will eventually slow down and stop
如果钟摆有摩擦力 最终它将减速并停止
and this is shown in phase space by the inward spiral —
这在相空间里通过向内螺旋线显示
the pendulum swings slower and less far each time
钟摆每次摆动都会更慢 摆动幅度更小
and it doesn’t really matter what the initial conditions are,
它的初始条件并不重要
we know that the final state will be the pendulum at rest hanging straight down
我们知道 最终的状态将会是钟摆静止垂直
and from the graph it looks like the system is attracted
从图上看 系统好像是被吸引
to the origin, that one fixed point
到原点这个固定点
so this is called a fixed point attractor
所以这被称为不动点吸引子
now if the pendulum doesn’t lose energy,
如果钟摆没有损失能量
well it swings back and forth the same way each time
它每次会以相同的方式来回摆动
and in phase space we get a loop
在相空间中 我们得到一个环
the pendulum is going fastest at the bottom
钟摆在底部的速度最快
but the swing is in opposite directions as it goes back and forth
但来回摆动的方向相反
the closed loop tells us the motion is periodic and predictable
这个闭环告诉我们运动是周期性的 是可预测的
anytime you see an image like this in phase space,
任何时候在相空间中看到这样的图像
you know that this system regularly repeats
就知道这个系统是在有规律地重复着
we can swing the pendulum with different amplitudes,
我们可以摆动具有不同振幅的钟摆
but the picture in phase space is very similar, just a different sized loop
但相空间中的图像和之前是相似的 只是环的大小不同
now an important thing to note is that
现在需要注意的重要一点是
the curves never cross in phase space
曲线在相空间中永远不会相交
and that’s because each point uniquely identifies
这是因为每个点都唯一地标识了
the complete state of the system
系统的完整状态
and that state has only one future
并且这个状态只有一种未来
so once you’ve defined the initial state,
所以一旦你定义了初始状态
the entire future is determined
整个未来就确定了
now the pendulum can be well understood using Newtonian physics,
用牛顿物理学可以很好地理解钟摆
but Newton himself was aware of problems
但牛顿自己知道问题
that did not submit to his equations so easily,
并不会那么容易遵循他的方程式
particularly the three-body problem.
尤其是三体问题
so calculating the motion of the Earth around the Sun was simple enough
因此 计算地球围绕太阳的运动很简单
with just those two bodies
因为只有这两个天体
but add in one more, say the moon,
但再加上一个天体 比如月球
and it became virtually impossible
它就几乎无法计算了
Newton told his friend Haley
牛顿告诉他的朋友哈利
that the theory of the motions of the moon made his head ache,
关于月球运动的理论让他很头疼
and kept him awake so often that he would think of it no more
并且使他经常睡不着觉 以至于不再去思考它了
the problem, as would become clear to Henri Poincaré two hundred years later,
两百年后 亨利·庞加莱清楚地认识到
was that there was no simple solution to the three-body problem
三体问题没有简单的解决方案
Poincaré had glimpsed what later became known as chaos.
庞加莱窥见了后来著名的混沌现象
Chaos really came into focus in the 1960s,
混沌在20世纪60年代成为焦点
when meteorologist Ed Lorenz tried to
当气象学家艾德·洛伦兹尝试
make a basic computer simulation of the Earth’s atmosphere
对地球大气层进行一个基本的计算机模拟时
he had 12 equations and 12 variables,
他用了12个方程 12种变量
things like temperature, pressure, humidity and so on
比如温度 压力 湿度等等
and the computer would print out each time
计算机会把每一个时间
step as a row of 12 numbers
步长打印成12个数字一行
so you could watch how they evolved over time
这样你就能观察它们是如何随着时间演变的
now the breakthrough came when Lorenz wanted to redo a run
突破发生在洛伦兹想要重新运行一遍时
but as a shortcut he entered the numbers from halfway
他依照一种快捷方式 在前一次打印的过程中
through a previous printout
输入了数字
and then he set the computer calculating
然后让电脑计算
he went off to get some coffee,
自己出去喝了咖啡
and when he came back and saw the results,
当他回来看到计算结果时
Lorenz was stunned.
洛伦兹惊呆了
The new run followed the old one for a short
新的数字跟随旧的数字运行了一小会
while but then it diverged
但之后就产生了偏离
and pretty soon it was describing
很快它描述了
a totally different state of the atmosphere
一种完全不同的大气状态
I mean totally different weather
我的意思是完全不同的天气
Lorenz’s first thought,
洛伦兹的第一反应
of course, was that the computer had broken
肯定是电脑坏了
Maybe a vacuum tube had blown.
可能是真空管爆了
But none had.
但没有
The real reason for the difference came down to the fact
造成这种差异的真正原因是
that printer rounded to three decimal places
打印机四舍五入到小数点后三位
whereas the computer calculated with six
而计算机用小数点后六位进行计算
So when he entered those initial conditions,
所以当他输入那些初始条件时
the difference of less than one part in a thousand
不到千分之一的差异
created totally different weather just a short time into the future
在未来很短的时间内创造了完全不同的天气
now Lorenz tried simplifying his equations
洛伦兹试着简化他的方程
and then simplifying them some more,
然后再进一步简化
down to just three equations and three variables
只剩下三个方程和三种变量
which represented a toy model of convection:
它们代表了一种对流模型:
essentially a 2d slice of the atmosphere heated at the bottom and cooled at the top
本质上是一个在底部加热 顶部冷却的二维切片
but again, he got the same type of behavior:
但再次 他得到了相同的反应:
if he changed the numbers just a tiny bit,
如果他稍微改变一下数字
results diverged dramatically.
结果就会大相径庭
Lorenz’s system displayed what’s become known as
洛伦兹的系统表现出所谓的
sensitive dependence on initial conditions,
对初始条件的敏感性
which is the hallmark of chaos
这就是混沌的标志
now since Lorenz was working with three variables,
既然洛伦兹用了三个变量
we can plot the phase space of his system in three dimensions
我们可以描绘出他的系统在三个维度上的相空间
We can pick any point as our initial state
我们可以选择任意一点作为初始状态
and watch how it evolves.
然后观察它怎样演化
Does our point move toward a fixed attractor?
我们的点是否会向一个固定吸引子移动?
Or a repeating loop?
还是重复循环?
It doesn’t seem to
似乎不会
In truth, our system will never revisit the same exact state again.
事实上 我们的系统再也不会重复同样的状态
Here I actually started with three closely spaced initial states,
在这里 实际上我是从三个很接近的初始状态开始
and they’ve been evolving together so far,
目前它们还是一起演化的
but now they’re starting to diverge
但现在开始分化了
From being arbitrarily close together,
从任意地接近开始
they end up on totally different trajectories.
他们最终走上了完全不同的轨迹
This is sensitive dependence on initial conditions in action.
这就是初始条件的敏感性依赖导致的
Now I should point out that
我现在要指出
there is nothing random at all about this system of equations.
这个方程组里完全没有随机性
It’s completely deterministic, just like the pendulum
就像钟摆一样 是完全确定的
so if you could input exactly the same initial conditions
所以如果你能够输入完全一样的初始条件
you would get exactly the same result
是会得到完全相同的结果
the problem is, unlike the pendulum, this system is chaotic
问题是 和钟摆不同 这个系统是混沌的
so any difference in initial conditions, no matter how tiny,
所以初始条件中 无论是多么小的差异
will be amplified to a totally different final state
都会被放大成完全不同的最终状态
It seems like a paradox,
这似乎是一种悖论
but this system is both deterministic and unpredictable
但这个系统同时是确定和不可预测的
because in practice, you could never know
因为在实践中 你不可能
the initial conditions with perfect accuracy,
完全准确地知道初始条件
and I’m talking infinite decimal places.
我说的是无限小数位
But the result suggests why even today with huge supercomputers,
但这一结果表明 即使如今有了大型的超级计算机
it’s so hard to forecast the weather more than a week in advance
依然很难提前一个多星期预测天气
In fact, studies have shown that
事实上 研究表明
by the eighth day of a long-range forecast,
在一个长期预测的第八天
the prediction is less accurate than
这个预测的准确性要比
if you just took the historical average conditions for that day
简单取当天的历史平均情况还低
and knowing about chaos,
气象学家了解混沌后
meteorologists no longer make just a single forecast
不再只是做单一的预测
instead they make ensemble forecasts,
而是进行整体预测
varying initial conditions and model parameters
使用不同的初始条件和模型参数
to create a set of predictions.
进行一系列的预测
Now far from being the exception to the rule,
现在 混沌的系统远非规则的例外
chaotic systems have been turning up everywhere.
它无处不在
The double pendulum, just two simple pendulums connected together, is chaotic
由两个钟摆简单连接在一起的双摆就是混沌的
here two double pendulums have been released simultaneously
在这里 两个双摆在几乎相同的初始条件下
with almost the same initial conditions
同时被释放
but no matter how hard you try,
但不论你多努力
you could never release a double pendulum
你不可能释放一个双摆
and make it behave the same way twice.
让它以同样的方式运动两次
its motion will forever be unpredictable
钟摆的运动是永远不可预测
you might think that chaos always requires
你可能会认为混沌一直需要
a lot of energy or irregular motions,
很多能量或是无规则运动
but this system of five fidgets spinners
但是这个由旋转臂上带有排斥磁铁的
with repelling magnets in each of their arms is chaotic too
五个解压旋转器组成的系统也是混沌的
At first glance the system seems to repeat regularly,
乍一看 这个系统似乎在有规律地重复
but if you watch more closely, you’ll notice some strange motions
但如果你近一点看 就会注意到一些奇怪的运动
a spinner suddenly flips the other way
一个旋转器突然转到另一个方向
Even our solar system is not predictable
甚至我们的太阳系也不可预测
a study simulating our solar system for a hundred million years into the future
一项模拟太阳系未来一亿年的研究发现
found its behavior as a whole to be chaotic
整个太阳系的运行是混沌的
with a characteristic time of about four million years
其特征时间大约为四百万年
that means within say 10 or 15 million years,
这意味着在1000万或1500万年内
some planets or moons may have collided
一些行星或卫星可能已经发生碰撞
or been flung out of the solar system entirely.
或是被完全抛出太阳系
The very system we think of as the model of order,
我们视为秩序模型的系统
is unpredictable on even modest timescales
即使在适度的时间尺度上 也是不可预测的
So how well can we predict the future?
那么我们能在多大程度上预测未来呢?
Not very well at all at least when it comes to chaotic systems
至少当涉及到混沌的系统时 这一点都不好说
The further into the future you try to predict the harder it becomes
你试图预测的未来越远 它就越难预测
and past a certain point,
并且超过了某个点
predictions are no better than guesses.
预测并不比猜测好多少
The same is true when looking into the past of chaotic systems
研究混沌系统的过去
and trying to identify initial causes
试着确定初始因素也是一样
I think of it kind of like a fog that
我觉得这就像一团
sets in the further we try to look into the future or into the past
我们越是试图更进一步地展望未来或过去越是浓的迷雾
Chaos puts fundamental limits on what we can know about the future of systems
混沌从根本上限制了我们对系统未来的了解
and what we can say about their past
以及对它过去的看法
But there is a silver lining
但还是有一线希望
Let’s look again at the phase space of Lorenz’s equations
让我们来看一下洛伦兹方程的相空间
If we start with a whole bunch of different initial conditions
如果我们开始时采用一些不同的初始条件
and watch them evolve,
然后观察它们的演化
initially the motion is messy.
一开始运动是混乱的
But soon all the points have moved towards or onto an object
但很快 所有的点开始向一个物体移动或移到一个物体上
the object, coincidentally, looks a bit like a butterfly.
巧合地是 这个物体看起来像一只蝴蝶
it is the attractor
它就是吸引子
For a large range of initial conditions,
对于大量的初始条件
the system evolves into a state on this attractor
系统会演变成这个吸引子上的一种状态
Now remember: all the paths traced out here never cross
记住:所有的的路径永不相交
and they never connect to form a loop,
也不会连接成环
If they did then they would continue on that loop forever
如果出现了相交或是环 那么它们就会一直循环下去
and the behavior would be periodic and predictable
行为就是周期性的和可测性的
so each path here is actually an infinite curve in a finite space.
所以这里的每条路径实际上是有限空间中的无限曲线
But how is that possible?
但这怎么可能呢?
Fractals. But that’s a story for another video
分形 但这又是另一个视频的故事了
this particular attractor is called the Lorenz attractor,
这种特定的吸引子被称为洛伦兹吸引子
Probably the most famous example of a chaotic attractor
这可能是混沌吸引子最著名的例子
though many others have been found for other systems of equations
尽管在其他方程组中也发现了很多其他的例子
now if people have heard anything about the butterfly effect,
如果人们听说过什么关于蝴蝶效应的事情
it’s usually about how tiny causes make the future unpredictable
那它通常是关于微小的事物如何引发不可预测的未来
but the science behind the butterfly effect also reveals
但是蝴蝶效应背后的科学也揭示了
a deep and beautiful structure underlying the dynamics
隐藏在动力学背后 一个能给系统运行提供有效见解的
one that can provide useful insights into the behavior of a system
深刻而美丽的动态结构
So you can’t predict how any individual state will evolve,
所以你无法预测任何单一状态将如何演化
but you can say how a collection of states evolves
但你可以说一组状态是如何演化
and, at least in the case of Lorenz’s equations,
并且 至少在洛伦兹方程的条件下
they take the shape of a butterfly
这些状态会形成蝴蝶的形状
Hey this part of the video is sponsored by LastPass,
本视频由LastPass部分赞助播出
the password manager with unlimited password storage
这个密码管理器拥有无限量的密码存储
and free cross-device sync
和免费的跨设备同步
before I used a password manager, I’ve got to admit,
我不得不承认 在我使用密码管理器之前
I used the same password for a lot of different accounts
我对许多不同的账户使用了相同的密码
and I know that is incredibly dangerous
我知道这非常危险
because if even one of those sites got hacked,
因为如果其中一个网站被黑了
then all of my important accounts would be exposed
那么我所有的重要账户都会被曝光
Quite the butterfly effect.
这就是个蝴蝶效应
LastPass auto generates strong passwords for you
LastPass会自动为你生成复杂的密码
so you have a different, indecipherable code for each website
这样你每个网站都会拥有独一无二 无法破解的密码
and let’s face it: if there’s anything in your life that you want to be chaotic,
让我们面对现实:如果你生活中有什么东西想变得混沌的
it is the characters in your passwords
那一定是密码字符
the best part is because it autofills user names and passwords on websites
最好的部分是因为它可以自动填写网站上的用户名和密码
or on iOS or Android apps and mobile sites,
或是填充iOS 安卓系统的app和网站上的用户名和密码
you never have to remember another password again. Not for the rest of your life
你一辈子都不用再记住另外的密码了
and that means no more writing down passwords,
这意味着你不用再写下密码
no more getting locked out of accounts,
不用再被锁定账户
no more password resets.
不用再重置密码
You can use your brain for what it’s meant to be doing.
你可以让你的大脑做该它应该做的事情
and if you want extra features like advanced multi-factor authentication,
如果你需要高级的多因素身份认证等额外功能
you can upgrade to LastPass premium
你可以升级到LastPass高级版
I got to say, it works like magic.
我不得不说 它像魔法一样有效
so put your passwords on autopilot with LastPass
所以 把你的密码放到LastPass上
Click the link below to find out more,
点击下方的链接发现更多功能吧
and thanks to LastPass for sponsoring this part of the video
感谢LastPass对此视频的部分赞助

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视频概述

蝴蝶效应究竟是什么原理?背后的科学是怎样的?一起来看看

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审核员SR

视频来源

https://www.youtube.com/watch?v=fDek6cYijxI

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