What Is Entropy, Really?

究竟什么是熵？

Entropy is one of the most maligned and misunderstood concepts in science. Maybe you’ve heard it defined as the “amount of disorder” in a system. And the second law of thermodynamics says the entropy of a closed system always increases over time. So you might think, why should you clean up your office if it will only get messier?

熵是科学界中最被诋毁和误解的概念之一。你可能听说过它被定义为系统中的“无序度”。热力学第二定律指出，封闭系统的熵总是随时间增加。所以你可能会想，既然办公室只会变得越来越乱，那为什么还要打扫呢？

That might be true, but you can’t blame it on entropy. The messy-room metaphor is often used to introduce the idea (it’s usually a teenager’s bedroom—can you relate?), but it’s misleading. See, disorder doesn’t mean messiness or chaos; it refers to the number of ways the parts of a system can be arranged without changing the overall state of the system.

这或许没错，但你不能把锅甩给熵。凌乱房间的比喻常被用来引入这个概念（通常是青少年的卧室——你能感同身受吗？），但这具有误导性。要知道，无序并不意味着脏乱或混乱；它指的是在不改变系统整体状态的前提下，系统各部分可以排列组合的方式数量。

For example, say your “system” is just a box full of air. Inside, at the microscopic level, the gas molecules are bouncing around like bumper cars. Now, imagine you could map the location and velocity of each particle at a given instant. That would be one possible arrangement, or microstate, but there are an infinitude of others, and they’re changing trillions of times a second.

例如，假设你的“系统”只是一个装满空气的盒子。在微观层面，气体分子像碰碰车一样四处碰撞。现在，想象一下你能绘制出某一瞬间每个粒子的位置和速度。这是一种可能的排列方式，即“微观状态”，但还有无数种其他的排列方式，而且它们每秒钟都在发生数万亿次的变化。

Of course, you can’t really see this stuff. Instead, what you observe are overall, macro-level properties like air pressure; if you sealed the box at sea level, that would be 14.7 pounds per square inch. And unless you add energy to the system, say by heating it, that doesn’t change. So all those microstates correspond to the macrostate of 14.7 psi. Get it?

当然，你无法真正看到这些。相反，你观察到的是整体的、宏观层面的属性，比如气压；如果你在海平面密封这个盒子，气压就是每平方英寸 14.7 磅。除非你向系统添加能量（比如加热），否则这个数值不会改变。因此，所有那些微观状态都对应着 14.7 psi 的宏观状态。明白了吗？

In other words, entropy is all about the link between the invisible atomic realm and the visible, measurable realm of objects, the world we inhabit. You could say it’s a conceptual and mathematical bridge between two levels of reality. I mean, c’mon, that’s pretty cool.

换句话说，熵的核心在于连接不可见的原子领域与可见、可测量的物体领域——也就是我们所处的世界。你可以说它是连接两个现实层面的概念和数学桥梁。说真的，这难道不酷吗？

Now, out of all possible outcomes, which ones actually occur? That’s basically random, so it’s a matter of probability. In fact, probability is fundamental to the idea of entropy, and this is what the messy-room image fails to capture. So I’m going to use a different analogy: rolling dice. Einstein once said “God doesn’t play dice with the universe.” Let’s just see about that, shall we?

那么，在所有可能的结果中，哪些会真正发生呢？这基本上是随机的，所以这是一个概率问题。事实上，概率是熵概念的基础，而这正是“凌乱房间”的比喻所无法体现的。因此，我将使用另一个类比：掷骰子。爱因斯坦曾说：“上帝不掷骰子。”让我们来看看事实是否如此，好吗？

Rolling the Bones / 掷骰子

Imagine you roll a six-sided game die. You get a number from 1 to 6, right? There are six possible outcomes, or states. If you roll the 20-sided die in Dungeons & Dragons, there are 20 possible states. If you want to wow your D&D pals, you could casually remark that this die has a higher entropy—because it has more possible outcomes.

想象一下你掷一颗六面骰子。你会得到 1 到 6 之间的一个数字，对吧？这里有六种可能的结果，即状态。如果你掷《龙与地下城》（D&D）中的 20 面骰子，就有 20 种可能的状态。如果你想让你的 D&D 伙伴们惊叹，你可以随口说这颗骰子的熵更高——因为它有更多可能的结果。

Now say you’re determining a character’s abilities in D&D, and you roll three six-sided dice. The three values can add up to anything between 3 and 18, but the various sums are not equally likely. For maximum dexterity, say, you need an 18. Well, there’s only one way to achieve that: Each die must come up a 6.

现在假设你正在确定 D&D 中角色的能力值，你掷了三颗六面骰子。这三个数值之和可以在 3 到 18 之间，但各种总和出现的概率并不相等。比如，为了获得最高敏捷度，你需要 18 点。好吧，实现这一点的唯一方法是：每颗骰子都必须掷出 6。

But if moderate dexterity is enough for you, you might be fine with a level 10. That’s easier to get, because there are more combinations that add up to 10—six unique sets of numbers to be exact.

但如果中等敏捷度对你来说就足够了，那么 10 点可能也行。这更容易实现，因为有更多的组合可以加起来等于 10——确切地说是六组独特的数字组合。

And if we roll the dice one at a time and take sequence into account, there are even more permutations. Take 6-3-1 on the left. You could get the same three values in five other ways: 1-6-3, 3-1-6, 3-6-1, 6-1-3, 6-3-1. (Yes, these are different outcomes because time’s arrow moves in one direction.) All in all, there’s 27 different ways to roll a 10.

如果我们一颗一颗地掷骰子并考虑顺序，排列方式就更多了。以左侧的 6-3-1 为例，你可以通过其他五种方式得到同样的三个数值：1-6-3、3-1-6、3-6-1、6-1-3、6-3-1。（是的，这些是不同的结果，因为时间之箭是单向流动的。）总而言之，掷出 10 点共有 27 种不同的方式。

If we look at all possible results for three dice, there are 216 distinct microstates. But what matters for the game is the sum of the three values—that’s our macrostate. So the odds of rolling an 18 are 0.4 percent (1 out of 216), while the odds of rolling a 10 are 12.5 percent (27 out of 216).

如果我们观察三颗骰子的所有可能结果，共有 216 种不同的微观状态。但对游戏而言，重要的是三个数值的总和——这就是我们的宏观状态。因此，掷出 18 点的概率是 0.4%（216 分之 1），而掷出 10 点的概率是 12.5%（216 分之 27）。

We can say the 10 state has a higher entropy because there are more ways it can occur. And because there are more ways it can occur, it’s more likely to occur. See? There’s no mysterious force increasing the entropy of a system. It’s just that states with higher entropy have a higher probability. It’s actually kinda simple, really.

我们可以说 10 点的状态具有更高的熵，因为它出现的途径更多。正因为它出现的途径更多，所以它发生的可能性也更大。明白了吗？并没有什么神秘的力量在增加系统的熵。这仅仅是因为熵更高的状态具有更高的概率。其实，这真的很简单。

Opposite World / 相反的世界

Now let’s think about this in terms of energy. Say you take a glass of cold water with a temperature of 50 degrees Fahrenheit, and you drop a hot, 120-degree ball of copper into it. What happens? Well, from experience, you’d expect the water to get warmer and the ball to get cooler, until they equalize at some temperature between 50 and 120 degrees. That’s called thermal equilibrium.

现在让我们从能量的角度来思考。假设你拿一杯 50 华氏度的冷水，然后丢进一个 120 度的热铜球。会发生什么？根据经验，你会预期水变暖，球变凉，直到它们在 50 到 120 度之间的某个温度达到平衡。这被称为热平衡。

But what do we mean when we say something gets warmer? We mean that its atoms and molecules increase in kinetic energy—they get more jiggly. Say the water gains 50 joules of thermal energy. Then, since energy is always conserved, we know that the copper ball cools off, losing the same 50 joules of energy.

但当我们说某物变暖时，是什么意思呢？意思是它的原子和分子的动能增加了——它们振动得更剧烈了。假设水获得了 50 焦耳的热能。那么，由于能量总是守恒的，我们知道铜球会冷却下来，失去同样的 50 焦耳能量。

But wait. What if, on a particular Tuesday, you dropped the hot ball in the cold water and the ball got hotter, increasing in thermal energy by 10 joules, while the water lost 10 joules and got colder? Did you just break physics? Nope. Energy is still conserved. You might find this disturbing, but it could happen. Why? Entropy. It’s one possible distribution of energy—just an extremely unlikely one. Basically, you won the Lotto.

等等。如果在一个特定的周二，你把热球丢进冷水里，结果球变得更热了，热能增加了 10 焦耳，而水失去了 10 焦耳变得更冷了呢？你刚刚打破了物理定律吗？没有。能量依然守恒。你可能会觉得这很令人不安，但它确实可能发生。为什么？因为熵。这只是能量的一种可能分布——只是概率极低而已。基本上，你中了彩票。

An Object Lesson / 一个实例教学

Now imagine you have a tiny little solid. It’s so tiny, it has only three atoms. (Remember the three-dice analogy?) Quantum mechanics tells us that atoms can only have certain energy levels—just like a die can roll a 2 or a 3 but not a 2.5. The point is that if the total energy of these three atoms is 10 units, then as we saw, there are 27 ways those 10 units of energy can be distributed.

现在想象你有一个微小的固体。它非常小，只有三个原子。（还记得三颗骰子的类比吗？）量子力学告诉我们，原子只能处于特定的能级——就像骰子只能掷出 2 或 3，而不能掷出 2.5。重点是，如果这三个原子的总能量是 10 个单位，那么正如我们所见，这 10 个单位的能量有 27 种分配方式。

Let’s take this just one step further: Say we have two tiny objects, A and B, with different amounts of thermal energy. Object A consists of two atoms (dice) and has a total energy of 3 units. B has three atoms (dice) and 7 units of energy. This puts the total energy in the system (A and B) at 10 units.

让我们再深入一步：假设我们有两个微小的物体 A 和 B，它们具有不同数量的热能。物体 A 由两个原子（骰子）组成，总能量为 3 个单位。B 有三个原子（骰子），能量为 7 个单位。这使得系统（A 和 B）的总能量为 10 个单位。