Physicists like to say that there is no law more sacred than the second law of thermodynamics: the entropy of a closed system always increases. But what does it mean? We’ll work through this slowly, but we have to define some terms first, stick around.

Microstates and macrostates

If your friend asks you for the time, you’re likely to round to the nearest quarter or five-minute interval instead of replying with the exact time down to the millisecond. For instance, your mind may automatically assign the times of 12:14, 12:15, 12:16 to the phrase quarter past twelve. Why do we do this? Because for most cases, a general idea of the current time is simply more useful than knowledge of the exact microsecond. In physics, we distinguish between these two levels of description. The less exact, but more useful, times (e.g. 12:15, 12:30, 12:45) are called macrostates and the more exact times, like what a stopwatch tracks, are called microstates (e.g. 12:01:024 and 12:02:130).

The entropy equation

Here’s another example: suppose you have a box with $1,000$ coins inside. You vigorously shake the box and let out one coin at a time writing down the resulting sequence of heads and tails. Each sequence — e.g. $HTTHHT$ — constitutes a microstate while a coarse-grained quantity, such as the total number of heads $\#H$, forms a macrostate. Entropy¹ is simply a way to count how many full-detail microstates belong to each coarse-grained macrostate. More precisely, if $M$ is a macrostate,

$$\underbrace{S(M)}_{\textcolor{gray}{\mathclap{\footnotesize\text{entropy}}}} \;=\; \log \overbrace{\Omega(M)}^{\textcolor{gray}{\mathclap{\footnotesize\text{number of microstates}}}}$$

If logarithms are a distant memory, don’t worry. The log is just a convenient way of writing astronomically large counts as manageable numbers; all you need for this essay is that the more microstates a macrostate contains, the higher its entropy.

Majority rule

Even though each sequence of heads and tails is equally likely, there is only one single sequence that produces exactly $1,000$ heads. With only one possibility, the entropy is as low as it gets: $S(\#H = 1000) = \log 1 = 0$.
On the other hand, there are more sequences with $500$ heads than there are atoms in the entire observable universe! aside The number of ways to get $500$ heads from our box is $\binom{1000}{500} \approx 10^{299}$ giving an entropy of $S(\#H = 500) \approx 299 \log 10$. There are only about $10^{80}$ atoms in the entire observable universe. Therefore, after shaking the box, our system naturally favors the outcome with the most possiblities and on-average we see sequences with close to $500$ heads. This is the essence of the second law.

The arrow of time

Much like our coins, the atoms in a room are constantly being jiggled around by thermal fluctuations, and there are far more ways to arrange them evenly across the room than crammed into one corner. This is why smoke released in a corner spreads until it fills the whole room, almost every rearrangement produces that configuration. The second law of thermodynamics is simply this observation, that physical systems drift toward their most likely state. It is also why entropy is often called a measure of disorder: the disordered arrangements are usually, but not always, the ones that can be realized in the most ways.

Now, record a video of the smoke starting from the corner of the room and play the movie backwards. A clip of two air molecules colliding like two billiard balls looks perfectly fine in reverse; the microscopic laws of physics are symmetric under the reversal of time. Yet, the smoke gathering itself back into one corner of the room looks absurd, and you would spot it in a second. Where does the difference come from? Not from the laws of motion, but from the counting we just did. There are overwhelmingly more ways for smoke to fill a room than to huddle in a corner, so the direction we call forward in time is simply the direction in which arrangements become more likely. The arrow of time emerges from statistics.²

Entropy is counting

I hope entropy now feels a little less mysterious. In essence, it is counting. Shake anything, a box of coins, a room full of air, the universe, and it drifts toward the arrangement with the most possibilities. But this leaves a puzzle. If entropy always increases, why doesn’t every ice cube fly apart into vapor to boost the entropy of the universe? Why do solids exist at all? That question deserves its own essay: Why does ice melt?

Footnotes

1. We assume natural units $k_B = 1$, so entropy is just a number. ↩

2. From the point of view of statistical physics, time is an emergent phenomenon; it arises from considering collections of particles. From the perspective of relativity, time is not emergent, it is baked directly into the theory through spacetime. ↩