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物理学中最被误解的概念

引言

这段视频要探讨的是物理学中最重要却最不为人所理解的概念之一。 它支配着从分子碰撞到巨大风暴的一切过程，也支配着从宇宙初始到它的整个演化，再到它不可避免的终结。事实上，它可能决定了时间的方向， 甚至是生命存在的原因。 要想感受这个话题的复杂性， 你只需要问一个简单的问题：地球从太阳那里得到了什么？

主持人："地球从太阳那里得到了什么？"

路人A："嗯，是光线。"

路人B："热量。"

路人C："温暖，光。"

路人D："维生素D？"

主持人："你是从紫外线得到维生素D的吗？"

路人E："嗯，很多能量。"

路人F："地球从这里得到能量。"

路人G："对，是能量。"

路人H："能量。"

主持人："说对了。"

主持人："每天，地球从太阳那里得到一定量的能量。那么相对于从太阳得到的这些能量，地球向太空辐射回去多少能量呢？"

路人A："可能没那么多。你知道，我不相信。它就是直接辐射回去。"

路人F："我说更少。"

路人H："更少。"

路人C："更少。我觉得更少。"

路人I："我猜大约70%？"

路人D："我觉得是一小部分。"

路人G："我说20%。"

路人B："因为我们使用了一些。"

主持人："我们使用了一些能量？"

路人E："我们消耗了很多，对吧？"

主持人："但是关于能量，它永远不会凭空消失。你没法真正‘用掉’能量。"

路人J："它必须是平衡的，不是吗？相同的量，对。"

路人K："你知道，因果关系。它在某些方面应该是相等的，对吧？"

主持人："在地球历史的大部分时间里，从太阳获得的能量应该和地球向太空辐射的能量完全相同。"

路人D："哇。"

主持人："因为如果我们不这样做，那么地球会变得更热。那将是一个问题。"

路人J："那将是一个大问题。"

主持人："所以，如果是这样的话...那我们究竟从太阳那里得到了什么？"

 路人F："这是个好问题。"

路人B："它让我们晒出漂亮的古铜色。"

主持人："它让我们晒出漂亮的古铜色，我喜欢。"

主持人："我们从太阳那里得到了一些特别的东西。"

路人J："我不知道。如果没有能量我们得到什么？"

主持人："但是没人谈论这个。"

要回答这个问题，我们需要回到两个世纪前的一个发现。

卡诺的理想热机

1813年冬天，法国正在被奥地利、普鲁士和俄罗斯的军队入侵。拿破仑一位将军的儿子萨迪·卡诺——那时还是个17岁的学生。1813年12月29日，他给拿破仑写了一封信，请求参战。然而，当时拿破仑战事缠身，没回这封信。几个月后，巴黎遭到攻击，卡诺如愿以偿：他随学生们在巴黎东边的一座城堡抵抗敌军。可他们远不如来犯的军队强大，巴黎在仅仅一天的战斗后就陷落了。卡诺被迫撤退，心灰意冷。

七年后，他去探望他的父亲，他父亲在拿破仑失势后逃往了普鲁士。他的父亲不仅是一位将军，还是一位物理学家。他写过一篇关于能量在机械系统中如何最有效传递的论文。当他的儿子来访时，他们长谈当时的重大突破：蒸汽机。蒸汽机当时已经被用于驱动船只、开采矿石以及挖掘港口。显然，未来国家的工业和军事力量取决于拥有最好的蒸汽机。但法国的设计正在落后于英国等其他国家。于是，萨迪·卡诺决心找出原因。

当时，即使是最好的蒸汽机也只能将热能的3%转换为有用的机械功。如果他能提升这个数字，就能让法国在世界舞台上重新获得巨大优势。于是他花了接下来的三年研究热机（热能发动机）。

他的一个关键见解涉及“理想热机”——即没有摩擦、没有能量损失的理想状态。

它大概是这样工作的：

取两根很大的金属棒，一根热的和一根冷的。这台热机由一个充满空气的腔室组成，热量只能通过腔室底部流进或流出。腔室内有一个活塞，它与飞轮相连。

腔室内的空气初始温度刚好低于热棒的温度。首先，热棒与腔室接触在一起，热量流入腔室并维持空气温度不变，空气膨胀推升活塞，带动飞轮转动。接下来，移开热棒，但腔室内的空气继续膨胀，只是这时因为没有热量进入，温度下降。在理想情况下，直到达到冷棒的温度。

接着，将冷棒与腔室接触，飞轮将活塞向下推。当空气被压缩时，热量传递到冷棒中。移开冷棒。飞轮进一步压缩气体，使其温度升高，直到刚好低于热棒的温度。然后再次连接热棒，循环重复。通过这个过程，热棒的热量被转换成飞轮的能量。

值得注意的是，卡诺的理想热机是完全可逆的。

如果你反向运行热机，首先空气膨胀降低温度，然后腔室与冷棒接触，空气进一步膨胀，从冷棒吸收热量。接下来，空气被压缩，温度升高。腔室放在热棒上方，飞轮的能量被用来将热量送回热棒。无论正向运行了多少个循环，你都可以反向运行相同的次数，最终，一切都会回到原始状态，而且不需要额外的能量输入。换言之，运行理想热机并不会真正改变什么，因为你总能把所做的一切再复原。

效率

那么这台热机的效率是多少？既然它是完全可逆的，你可能会认为效率是100%，但事实并非如此。

每次循环，飞轮获得的能量是从热棒流入腔室的热量减去从腔室流向冷棒的热量。所以要计算效率，我们用飞轮得到的能量除以从热棒输入的热量。

热棒一侧的输入热量等于气体对活塞所做的功，而这总是要大于活塞对气体所做的功（也就是冷棒侧的热量输出），这是因为同样的气体在温度较高时，相对于其温度较低时，对活塞施加的压力要更大。要提高发动机的效率，你可以提高热端的温度，或降低冷端的温度，或两者都做。

开尔文勋爵了解到卡诺的理想热机，并意识到它可以成为绝对温标的基础。想象一下，让气体膨胀到极限——冷却到所有气体分子几乎完全停止运动的程度。那样的话，气体将不再对活塞施加任何压力，在冷端压缩气体时就不需要做功，也就不会有热量损失。这就是绝对零度的概念，此时就能实现 100% 的热机效率。

借助这个绝对温标，也就是开尔文（Kelvin）温标，我们可以用热端和冷端的温度来代替热量的输入和输出，因为二者成正比。于是效率可以写成这样，并可以改写成这样：

我们由此得知，理想热机的效率不依赖于其内部材料或设计，而根本取决于热端和冷端的温度。要达到100%的效率，你需要热端有无限高的温度或冷端达到绝对零度，而这在现实中都是不可能的。即使没有摩擦和能量损耗，100% 效率也无法实现。其原因在于，要让活塞回到初始位置，就必须把部分热量排到冷棒中，所以并不能把全部能量都留在飞轮里。

在卡诺的时代，高压蒸汽机的温度只能达到大约 160°C，所以它们的理论最高效率大约是 32%，而实际效率只有 3% 左右。这是因为现实中的热机会有摩擦、会向环境散热，也无法在恒温下传热。结果就是，同样的热量输入，最终进入飞轮的能量更少，其余的能量会分散到气缸壁、飞轮轴以及被辐射到环境中。

当能量像这样扩散时，它是不可能被收回的。所以这个过程是不可逆的。总能量并没有改变，但它变得不再集中可用。能量在集中状态下最有用，而在分散状态下可用性就降低了。

熵

几十年后，德国物理学家鲁道夫·克劳修斯研究卡诺的热机，并提出了一种衡量能量扩散程度的方法。他称这个量为熵（entropy）。

当所有能量都集中在热棒中时，熵就处于低水平；但当能量向环境、腔室壁、转轴等处扩散时，熵就会增加。这意味着能量总量保持不变，但在这种更分散的形式下，它做功的能力降低了。

1865年，克劳修斯用这样的话概括了热力学前两定律：第一，宇宙的能量是恒定的。第二，宇宙的熵趋向于最大化。换句话说，能量会随着时间的推移而不断扩散。

第二定律是世界上许多现象的核心。它解释了为什么热的东西会变冷，冷的东西会变热，为什么气体会膨胀并占满整个容器，为什么不存在永动机——因为在一个封闭系统里，可用能量总是在减少。

能量扩散

“熵”最常见的描述方式是“无序度”，因为熵增大往往伴随着更混合、更随机、更无序的状态。但我认为思考熵的最佳方式是：能量随时间扩散的趋势。

那么为什么能量会随时间扩散呢？我是说，大多数物理定律在时间正向或反向运行时都完全相同。那么这种明显的时间依赖性是如何产生的？

让我们来考虑两个小金属棒，一个热的，一个冷的。为了便于讨论，我们先假设每个棒只有 8 个原子。每个原子根据它拥有的能量包数量而振动。能量包越多，振动越剧烈。现在先设定左边的棒拥有 7 个能量包，右边则有 3 个。每个棒各自拥有的能量包数量就是我们所说的系统状态。

首先，只考虑左边的棒，它有 7 个能量包。这些能量包可以随机地在晶格中跳动。这种随机跳动从不停歇。能量包随机地从一个原子跳到另一个原子，导致不同的能量分布（配置）。但总能量始终保持 7 个能量包不变。

现在重新把只有 3 个能量包的冷棒加进来并让它们接触。这样，能量包就可以在两个棒之间跳动，产生不同的组合配置。每种组合配置出现的可能性都是相等的。

那么如果我们在某一瞬间暂停，看看能量包都跑到了哪里。停！看看这个。现在左边金属棒有 9 个能量包，右边只有 1 个。所以热量从冷处流向了热处。这不是不应该发生吗？这不是会导致熵减吗？

这就是路德维希·玻尔兹曼的重要见解：热从冷流向热并非不可能，只是概率极低而已。

拥有 9 个能量包在左棒、1 个在右棒的组合方式有 91,520 种，而左棒和右棒各有 5 个能量包的组合方式有 627,264 种，也就是能量平均分布的方式比“左多右少”的情况概率高出 6 倍以上。但如果把所有可能性都加总起来，还是有 10.5% 的机会让左棒能量包比初始更多。

那么，为什么我们在周围观察不到这种现象呢？

好吧，看看当我们增加原子数量到每根金属棒 80 个，能量包增加到 100 个时会发生什么，其中左边 70 个，右边 30 个。现在左棒最终比初始更热（也就是比 70 个更多）的概率只有 0.05%。随着我们继续扩大系统规模，这个趋势会持续。比如日常生活中随便一个固体就包含大约 1026个原子，而能量包也远远多于这个数量，所以热量从冷处流向热处是如此不可能，以至于从不会发生。

你可以把它想成这个魔方。最初它是复原的，但我要闭上眼睛随机转动它。如果持续这样下去，魔方会被打乱得越来越厉害。但我怎么能确定自己真的是在“弄乱”它呢？因为魔方“完全复原”的状态只有唯一一种，“几乎复原”的状态也只有极少数，而“几乎完全随机”的状态有数以 quintillion（百京）计。没有思考和努力地操作下，每次转动都让魔方从一个极不可能的状态（复原）变成更常见的状态（打乱）。

空调

如果能量的自然趋势是不断向外扩散、使事物变得更无序，那么像空调这样让房子内部的冷空气变得更冷、而炎热的外部变得更热的现象又是如何实现的呢？

能量在从冷的一方流向热的一方，从而降低了房子的熵。然而，这种熵的降低只能通过在其他地方增加更多的熵来实现。

在这里，以燃煤发电厂为例，煤炭中集中的化学能被释放，发电厂及其环境被加热，然后能量传递到涡轮机和发电机，通过电线一路传到房屋，在风扇和压缩机的运作过程中又产生废热。房子内部获得的熵减少，远远被为了实现这一过程而在其他地方增加的熵所抵消。

但如果总熵在不断增加，并且我们做的一切都只会加速这种增加，那么地球上怎么还会存在任何结构？为什么还会有冷热分离的区域？生命又是如何产生并持续存在的？

好吧，如果地球是一个封闭系统，能量会完全扩散，意味着，所有生命都会终止，一切都会衰变混合，最终达到相同的温度。但幸运的是，地球不是一个封闭系统，因为我们有太阳。

太阳真正赋予我们的是持续不断的低熵能量流，也就是集中且束缚在一起的能量。

主持人“我们从太阳得到的能量比我们返回的能量更有用。它更紧凑，更聚集在一起。”

植物捕捉这股能量，用它来生长并制造糖分。然后动物吃植物，用那些能量来维持身体机能和活动。更大的动物通过吃小动物来获取能量，如此层层递进。每一步都使能量逐渐扩散。

路人："好，有意思。"

主持人："是的。"

路人："哦哇，我不知道这个。"

主持人："这就对了。"

最终，所有从太阳到达地球的能量都转换成了热能，然后再被辐射回太空。但实际上，这是相同的量。

主持人："我知道这不是..."

路人：“你，你知道这是...”

主持人："我是物理学博士。"

路人："哦好的。好的。 我相信你。"

地球生命

熵的增加可以从到达和离开地球的光子数量比例中看出来。地球每接收一个来自太阳的光子，就会发射出 20 个光子。

而地球上发生的一切，植物生长、树木倒下、兽群奔跑，飓风和龙卷风，人们吃喝、睡眠和呼吸等等，都是在把少量的高能量光子转化为 20 倍数量、能量更低的光子的过程中发生的。

如果没有一个集中的能量来源以及一种丢弃已扩散能量的方式，地球上的生命就不可能存在。

甚至有人认为生命本身可能是热力学第二定律的必然结果。如果宇宙趋向于最大熵，那么生命提供了一种加速这种自然趋势的方式，因为生命在将低熵转换为高熵方面表现得异常出色。

例如，当存在蓝藻和其他有机物时，海水表层产生的熵比没有时多 30%到 680%。

杰里米·英格兰更进一步提出，如果存在持续的高度集中的能量流，这种环境就会有利于那些能够消散能量的结构出现。随着时间推移，这些结构会演变得越来越擅长消散能量，最终产生生命。

用他自己的话说，"如果你从一团随机的原子开始，并且长时间用光照射它们，那么最终得到一株植物也就不足为奇了。"

所以地球上的生命靠太阳的低熵生存。

但太阳的低熵又是从哪里来的呢？

过去假说

答案在于整个宇宙。如果我们知道宇宙的总熵随时间增加，那么昨天的熵就比今天低，前天的熵比昨天还要低，如此一直回溯到宇宙大爆炸时。所以在大爆炸之后，那时熵是最低的。这被称作过去假说（past hypothesis）。它并不解释为什么熵很低，只是说明必须要有一个低熵的初始条件，宇宙才能演化成现在的样子。

但早期宇宙是炎热、稠密的，而且几乎完全均匀。我是说，一切都混合在一起，各处的温度几乎相同，相差最多不超过 0.001%。

那这怎么算是低熵呢？

关键在于我们还没有考虑引力。引力会使物质凝聚在一起。所以如果考虑了引力，把物质完全分散得如此均匀，其实是极不可能的状态，这也正是为什么当时的熵很低。

随着宇宙的膨胀和冷却，物质开始在更致密的区域凝聚。在这个过程中，巨量的势能转化为动能，这些能量也可以被利用。比如像水往低处流可以驱动涡轮机那样。但当物质彼此碰撞时，它们的一些动能会转化为热能，因而可用能量减少，熵随之增加。随着时间推移，可用能量被使用，恒星、行星、星系以及生命在此过程中形成，也不断提升了熵。

宇宙最初大约有 1088 个玻尔兹曼常数的熵。如今，可观测宇宙中的所有恒星大约有9.5乘以10的80次方。星际和星系际介质总共约有它的近 10 倍，但依旧只是当初宇宙的一小部分。中微子和宇宙微波背景辐射中的光子则包含了更多的熵。

霍金辐射

1972年，雅各布·贝肯斯坦提出了另一个熵的来源，黑洞。

他提出黑洞的熵应该与其表面积成正比。也就是说，黑洞越大，熵越高。

很多著名物理学家起初都觉得这个想法很荒谬，这也不无道理。根据经典热力学，如果黑洞有熵，那么它们也应该有温度。但如果有温度，就应该会辐射出能量，那它就不再“黑”了。当时打算证明贝肯斯坦是错的人正是斯蒂芬·霍金。

但令他惊讶的是，他的研究表明黑洞确实会发出辐射，这就是后来所说的霍金辐射，它们的确具有温度。

银河系中心的黑洞温度约为开尔文度的万亿分之一，发出的辐射微弱得无法探测。所以看起来依然很“黑”。但霍金证实了黑洞确实有熵，而贝肯斯坦是对的。霍金进一步完善贝肯斯坦的理论并确定它们具有多少熵。

比如，银河系中心的超大质量黑洞约有10的91次方玻尔兹曼常数的熵。相当于早期可观测宇宙熵的上千倍，也比所有其他粒子加在一起还要多 10 倍。而这仅仅是一个黑洞而已。所有黑洞加在一起的熵值为3乘以10的104次方玻尔兹曼常数。

也就是说，几乎整个宇宙的熵都集中在黑洞之中。换句话说，早期宇宙的熵只占了现在宇宙熵的大约 0.00000000000003%。

宇宙热寂

所以熵最初很低，而宇宙中发生的一切，比如行星系统的形成，星系的合并，小行星的碰撞，恒星的消亡，直到生命本身的繁荣，所有这些都能发生是因为宇宙的熵最初很低，并且一直在增加，而且这一切都只朝一个方向发展。我们从未见过小行星"反碰撞"，或行星系统"反混合"回到形成它的尘埃和气体云。过去和未来之间存在着明显的差异，而这种差异源于熵。

正是因为我们从一个“不大可能”的状态走向一个“更大可能”的状态，才形成了所谓的时间之箭。

预计这一过程会一直持续下去，直至能量彻底分散，以至于再也不会发生任何有趣的事情。

这就是宇宙的热寂。在遥远的未来，超过10的100次方年后，当最后一个黑洞也蒸发殆尽时，宇宙将处于它最可能出现的状态。那时，即使在宏观尺度上，你也无法分辨时间是在向前还是向后流动，时间之箭本身也将消失。

结论

听起来，好像熵是一个可怕的东西，必然会将我们引向可以想象到的最无聊的结果。但最大熵具有低复杂度，并不意味着低熵就拥有最高的复杂度。

实际上更像这杯茶和牛奶。你看，把茶杯这样拿着并不怎么吸引人，但当我倒入牛奶时，两者开始混合，就会产生非常漂亮的图案。它们瞬间出现，转眼之间，又消失得无影无踪。

低熵和高熵的复杂度都很低，恰恰是在它们中间的阶段，复杂的结构出现并繁盛。

既然我们正处于这个阶段，那就趁还有低熵可用的时候，充分加以利用吧。

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## The Arrow of Time

This is a video about one of the most important, yet least understood concepts in all of physics. It governs everything from molecular collisions to humongous storms, from the beginning of the universe through its entire evolution to its inevitable end. It may, in fact, determine the direction of time and even be the reason that life exists. To see the confusion around this topic, you need ask only one simple question: What does the Earth get from the sun?

### What Does Earth Get From the Sun?

(Street interviews with various answers including light rays, heat, warmth, light, Vitamin D, and energy)

Nailed it! Every day, the Earth gets a certain amount of energy from the sun. And then, how much energy does the Earth radiate back into space relative to that amount that it gets from the sun?

(Street interviews with various answers including less than it receives, 70%, a fraction, 20%, etc. with explanations like "because we use some of it.")

But the thing about energy is it never really goes away. You can't really use it up.

(Street interviews with answers including "it would have to break even," "same amount," "cause and effect, it would be equal in some ways.")

For most of the Earth's history, it should be exactly the same amount of energy in from the sun as Earth radiates into space.

(Street interviews with comments like "Wow!" and "Because if we didn't do that, then the Earth would get a lot hotter. That would be a problem.")

So, if that is the case, then what are we really getting from the sun?

(Street interviews with answers including "That's a good question," "It gives us a nice tan," and "I don't know. What do we get without energy?")

Nobody talks about it. To answer that, we have to go back to a discovery made two centuries ago.

### Sadi Carnot and the Ideal Heat Engine

In the winter of 1813, France was being invaded by the armies of Austria, Prussia, and Russia. The son of one of Napoleon's generals was Sadi Carnot, a 17-year-old student. On December 29th, he writes a letter to Napoleon to request to join in the fight. Napoleon, preoccupied in battle, never replies. But Carnot gets his wish a few months later when Paris is attacked. The students defend a chateau just east of the city, but they are no match for the advancing armies and Paris falls after only a day of fighting. Forced to retreat, Carnot is devastated.

Seven years later, he goes to visit his father who has fled to Prussia after Napoleon's downfall. His father was not only a general, but also a physicist. He wrote an essay on how energy is most efficiently transferred in mechanical systems. When his son comes to visit, they talk at length about the big breakthrough of the time: steam engines. Steam engines were already being used to power ships, mine ore, and excavate ports. And it was clear that the future industrial and military might of nations depended on having the best steam engines. But French designs were falling behind those of other countries, like Britain. So Sadi Carnot took it upon himself to figure out why.

At the time, even the best steam engines only converted around 3% of thermal energy into useful mechanical work. If he could improve on that, he could give France a huge advantage and restore its place in the world. So he spends the next three years studying heat engines. And one of his key insights involves how an ideal heat engine would work, one with no friction and no losses to the environment. It looks something like this:

Take two really big metal bars, one hot and one cold. The engine consists of a chamber filled with air where heat can only flow in or out through the bottom. Inside the chamber is a piston, which is connected to a flywheel. The air starts at a temperature just below that of the hot bar. So first, the hot bar is brought into contact with the chamber. The air inside expands with heat flowing into it to maintain its temperature. This pushes the piston up, turning the flywheel. Next, the hot bar is removed, but the air in the chamber continues to expand, except now without heat entering, the temperature decreases. In the ideal case, until it is the temperature of the cold bar.

The cold bar is brought into contact with the chamber and the flywheel pushes the piston down. And as the air is compressed, heat is transferred into the cold bar. The cold bar is removed, the flywheel compresses the gas further, increasing its temperature until it is just below that of the hot bar. Then the hot bar is connected again and the cycle repeats. Through this process, heat from the hot bar is converted into the energy of the flywheel.

And what's interesting to note about Carnot's ideal engine is that it is completely reversible. If you ran the engine in reverse, first the air expands, lowering the temperature. Then the chamber is brought into contact with the cold bar, the air expands more, drawing in heat from the cold bar. Next, the air is compressed, increasing its temperature. The chamber is placed on top of the hot bar, and the energy of the flywheel is used to return the heat back into the hot bar. However many cycles were run in the forward direction, you could run the same number in reverse, and at the end, everything would return to its original state with no additional input of energy required. So by running an ideal engine, nothing really changes. You can always undo what you did.

### Efficiency and Entropy

So what is the efficiency of this engine? Since it's fully reversible, you might expect the efficiency to be 100%. But that is not the case. Each cycle, the energy of the flywheel increases by the amount of heat flowing into the chamber from the hot bar, minus the heat flowing out of the chamber at the cold bar. So to calculate the efficiency, we divide this energy by the heat input from the hot bar.

Now the heat in on the hot side is equal to the work done by the gas on the piston. And this will always be greater than the work done by the piston on the gas on the cold side, which equals the heat out. And this is because on the hot side, the hot gas exerts a greater pressure on the piston than that same gas when cold. To increase the efficiency of the engine, you could increase the temperature of the hot side or decrease the temperature of the cold side, or both.

Lord Kelvin learns of Carnot's ideal heat engine and realizes it could form the basis for an absolute temperature scale. Imagine that the gas is allowed to expand an extreme amount, so much that it cools to the point where all the gas particles effectively stop moving. Then they would exert no pressure on the piston, and it would take no work to compress it on the cold side. So no heat would be lost. This is the idea of absolute zero, and it would make for a 100% efficient engine.

Using this absolute temperature scale, the Kelvin scale, we can replace the amount of heat in and out with the temperature of the hot and cold side respectively because they are directly proportional. So we can express efficiency like this, which we can rewrite like this.

What we have learned is that the efficiency of an ideal heat engine doesn't depend on the materials or the design of the engine, but fundamentally on the temperatures of the hot and cold sides. To reach 100% efficiency, you'd need infinite temperature on the hot side or absolute zero on the cold side, both of which are impossible in practice. So even with no friction or losses to the environment, it's impossible to make a heat engine 100% efficient. And that's because to return the piston to its original position, you need to dump heat into the cold bar. So not all the energy stays in the flywheel.

Now, in Carnot's time, high-pressure steam engines could only reach temperatures up to 160° Celsius. So their theoretical maximum efficiency was 32%, but their real efficiency was more like 3%. That's because real engines experience friction, dissipate heat to the environment, and they don't transfer heat at constant temperatures. So for just as much heat going in, less energy ends up in the flywheel. The rest is spread out over the walls of the cylinder, the axle of the flywheel, and is radiated out into the environment.

When energy spreads out like this, it is impossible to get it back. So this process is irreversible. The total amount of energy didn't change, but it became less usable. Energy is most usable when it is concentrated and less usable when it's spread out.

Decades later, German physicist Rudolf Clausius studies Carnot's engine, and he comes up with a way to measure how spread out the energy is. He calls this quantity entropy. When all the energy is concentrated in the hot bar, that is low entropy. But as the energy spreads to the surroundings, the walls of the chamber and the axle, well, entropy increases. This means the same amount of energy is present, but in this more dispersed form, it is less available to do work.

In 1865, Clausius summarizes the first two laws of thermodynamics like this: First, the energy of the universe is constant. And second, the entropy of the universe tends to a maximum. In other words, energy spreads out over time.

The second law is core to so many phenomena in the world. It's why hot things cool down and cool things heat up, why gas expands to fill a container, why you can't have a perpetual motion machine, because the amount of usable energy in a closed system is always decreasing.

The most common way to describe entropy is as disorder, which makes sense because it is associated with things becoming more mixed, random, and less ordered. But I think the best way to think about entropy is as the tendency of energy to spread out.

So why does energy spread out over time? I mean, most of the laws of physics work exactly the same way forwards or backwards in time. So how does this clear time dependence arise?

### Why Does Energy Spread Out?

Well, let's consider two small metal bars, one hot and one cold. For this simple model, we'll consider only eight atoms per bar. Each atom vibrates according to the number of energy packets it has. The more packets, the more it vibrates. So let's start with seven packets of energy in the left bar and three in the right. The number of energy packets in each bar is what we'll call a state.

First, let's consider just the left bar. It has seven energy packets, which are free to move around the lattice. This happens nonstop. The energy packets hop randomly from atom to atom, giving different configurations of energy. But the total energy stays the same the whole time.

Now let's bring the cold bar back in with only three packets and touch them together. The energy packets can now hop around between both bars, creating different configurations. Each unique configuration is equally likely. So what happens if we take a snapshot at one instant in time and see where all the energy packets are? So, stop.

Look at this. Now there are nine energy packets in the left bar and only one in the right bar. So heat has flowed from cold to hot. Shouldn't that be impossible because it decreases entropy? Well, this is where Ludwig Boltzmann made an important insight. Heat flowing from cold to hot is not impossible; it's just improbable.

There are 91,520 configurations with nine energy packets in the left bar, but 627,264 with five energy packets in each bar. That is, the energy is more than six times as likely to be evenly spread between the bars. But if you add up all the possibilities, you find there's still a 10.5% chance that the left bar ends up with more energy packets than it started.

So why don't we observe this happening around us? Well, watch what happens as we increase the number of atoms to 80 per bar and the energy packets to 100, with 70 in the left bar and 30 in the right. There is now only a 0.05% chance that the left solid ends up hotter than it started. And this trend continues as we keep scaling up the system. In everyday solids, there are around 100 trillion trillion atoms and even more energy packets. So heat flowing from cold to hot is just so unlikely that it never happens.

Think of it like this Rubik's Cube. Right now, it is completely solved. But I'm going to close my eyes and make some turns at random. If I keep doing this, it will get further and further from being solved. But how can I be confident that I'm really messing this cube up? Well, because there's only one way for it to be solved, a few ways for it to be almost solved, and quintillions of ways for it to be almost entirely random. Without thought and effort, every turn moves the Rubik's Cube from a highly unlikely state, that of it being solved, to a more likely state, a total mess.

### Entropy, Structure, and Life

So if the natural tendency of energy is to spread out and for things to get messier, then how is it possible to have something like air conditioning, where the cold interior of a house gets cooler and the hot exterior gets hotter? Energy is going from cold to hot, decreasing the entropy of the house.

Well, this decrease in entropy is only possible by increasing the entropy a greater amount somewhere else. In this case, at a power plant, the concentrated chemical energy in coal is being released, heating up the power plant and its environment, spreading to the turbine, the electric generators, heating the wires all the way to the house and producing waste heat in the fans and compressor. Whatever decrease in entropy is achieved at the house is more than paid for by an increase in entropy required to make that happen.

But if total entropy is constantly increasing, and anything we do only accelerates that increase, then how is there any structure left on Earth? How are there hot parts separate from cold parts? How does life exist?

Well, if the Earth were a closed system, the energy would spread out completely, meaning all life would cease, everything would decay and mix, and eventually reach the same temperature. But luckily, Earth is not a closed system because we have the sun.

What the sun really gives us is a steady stream of low entropy, that is concentrated, bundled up energy.

(Street interview: "The energy that we get from the sun is more useful than the energy we give back. It's more compact, it's more clumped together.")

Plants capture this energy and use it to grow and create sugars. Then animals eat plants and use that energy to maintain their bodies and move around. Bigger animals get their energy by eating smaller animals, and so on. And each step of the way, the energy becomes more spread out.

(Street interviews: "Okay, interesting," "Yeah," and "Oh wow, I did not know that.")

There you go. Ultimately, all the energy that reaches Earth from the sun is converted into thermal energy. And then it's radiated back into space.

(Street interview: "But in fact, it's the same amount." "You do know this is..." "I'm a Ph.D. physicist." "Okay, okay, I trust you.")

The increase in entropy can be seen in the relative number of photons arriving at and leaving the Earth. For each photon received from the sun, 20 photons are emitted. And everything that happens on Earth: plants growing, trees falling, herds stampeding, hurricanes and tornadoes, people eating, sleeping and breathing, all of it happens in the process of converting fewer, higher energy photons into 20 times as many lower energy photons.

Without a source of concentrated energy and a way to discard the spread-out energy, life on Earth would not be possible. It has even been suggested that life itself may be a consequence of the second law of thermodynamics. If the universe tends toward maximum entropy, then life offers a way to accelerate that natural tendency because life is spectacularly good at converting low entropy into high entropy.

For example, the surface layer of seawater produces between 30 to 680% more entropy when cyanobacteria and other organic matter is present than when it's not. Jeremy England takes this one step further. He's proposed that if there is a constant stream of clumped-up energy, this could favor structures that dissipate that energy. And over time, this results in better and better energy dissipators, eventually resulting in life. Or, in his own words, "You start with a random clump of atoms, and if you shine light on it for long enough, it should not be so surprising that you get a plant."

### The Sun, the Big Bang, and Black Holes

So life on Earth survives on the low entropy from the sun. But then where did the sun get its low entropy? The answer is the universe. If we know that the total entropy of the universe is increasing with time, then it was lower entropy yesterday and even lower entropy the day before that, and so on, all the way back to the Big Bang.

So right after the Big Bang, that is when the entropy was lowest. This is known as the past hypothesis. It doesn't explain why the entropy was low, just that it must have been that way for the universe to unfold as it has.

But the early universe was hot, dense, and almost completely uniform. I mean, everything was mixed, and the temperature was basically the same everywhere, varying by at most 0.001%. So how is this low entropy?

Well, the thing we've left out is gravity. Gravity tends to clump matter together. So taking gravity into account, having matter all spread out like this would be an extremely unlikely state. And that is why it's low entropy.

Over time, as the universe expanded and cooled, matter started to clump together in more dense regions. And in doing so, enormous amounts of potential energy were turned into kinetic energy. And this energy could also be used, like how water flowing downhill can power a turbine. But as bits of matter started hitting each other, some of their kinetic energy was converted into heat. So the amount of useful energy decreased, thereby increasing entropy. Over time, the useful energy was used. In doing so, stars, planets, galaxies, and life were formed, increasing entropy all along.

The universe started with around 10 to the 88th Boltzmann constants worth of entropy. Nowadays, all the stars in the observable universe have about 9.5 * 10 to the 80th. The interstellar and intergalactic medium combined have almost 10 times more, but still only a fraction of the early universe. A lot more is contained in neutrinos and in photons of the cosmic microwave background.

In 1972, Jacob Bekenstein proposed another source of entropy: black holes. He suggested that the entropy of a black hole should be proportional to its surface area. So as a black hole grows, its entropy increases. Famous physicists thought the idea was nonsense, and for good reason. According to classical thermodynamics, if black holes have entropy, then they should also have a temperature. But if they have temperatures, they should emit radiation and not be black after all.

The person who set out to prove Bekenstein wrong was Stephen Hawking. But to his surprise, his results showed that black holes do emit radiation, now known as Hawking radiation, and they do have a temperature. The black hole at the center of the Milky Way has a temperature of about a hundred trillionth of a Kelvin, emitting radiation that is far too weak to detect, so still pretty black. But Hawking confirmed that black holes have entropy, and Bekenstein was right. Hawking was able to refine Bekenstein's proposal and determine just how much entropy they have. The supermassive black hole at the center of the Milky Way has about 10 to the 91st Boltzmann constants of entropy. That is a thousand times as much as the early observable universe and 10 times more than all the other particles combined. And that is just one black hole. All black holes together account for 3 * 10 to the 104th Boltzmann constants worth of entropy.

So almost all the entropy of the universe is tied up in black holes. That means the early universe only had about 0.00000000000003% of the entropy it has now.

So the entropy was low, and everything that happens in the universe, like planetary systems forming, galaxies merging, asteroids crashing, stars dying, to life itself flourishing, all of that can happen because the entropy of the universe was low and it has been increasing. And it all happens only in one direction. We never see an asteroid un-crash or a planetary system un-mix into the cloud of dust and gas that made it up. There is a clear difference between going to the past and the future, and that difference comes from entropy. The fact that we are going from unlikely to more likely states is why there is an arrow of time.

### The Heat Death of the Universe

This is expected to continue until, eventually, the energy gets spread out so completely that nothing interesting will ever happen again. This is the heat death of the universe. In the distant future, more than 10 to the 100th years from now, after the last black hole has evaporated, the universe will be in its most probable state. Now, even on large scales, you would not be able to tell the difference between time moving forwards or backwards, and the arrow of time itself would disappear.

So it sounds like entropy is this awful thing that leads us inevitably towards the dullest outcome imaginable. But just because maximum entropy has low complexity does not mean that low entropy has maximum complexity. It's actually more like this tea and milk. I mean, holding it like this is not very interesting, but as I pour the milk in, the two start to mix and these beautiful patterns emerge. They arise in an instant, and before you know it, they're gone, back to being featureless. Both low and high entropy are low in complexity. It's in the middle where complex structures appear and thrive. And since that's where we find ourselves, let's make use of the low entropy we've got while we can.

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