Entropy

01Entropy is a count

Put "disorder" aside for a minute. Entropy counts something precise: how many ways the tiny parts of a thing can be arranged without changing how the thing looks from far away.

Physicists split every situation in two. The microstate is the full detail: which coin landed heads, where each molecule sits. The macrostate is the summary you'd actually notice: two heads out of four, a warm cup of coffee. Entropy measures how many microstates hide behind one macrostate.

"Two heads" wins because six of the sixteen arrangements belong to it, while "all heads" has exactly one. Every arrangement is equally likely; the columns differ only in how many arrangements they hold. Watch what that does once the coins pile up.

That's the second law of thermodynamics with the mystery taken out. A system wanders between its arrangements more or less at random, so it ends up in the macrostate with the most of them, because that's where nearly all the arrangements are. A glass of water holds about 10²⁵ molecules. At numbers like that, "overwhelmingly likely" is as close to "always" as anything in the universe gets.

02Open the gate

A box with a wall down the middle. Every particle starts on the left. Open the gate and watch the count.

Try 6 particles first. They slosh around, and every so often all six land back on the left by pure chance (the counter catches it). Switch to 300 and that never happens: the split hugs 50/50 and the wiggles shrink. Nothing pushes the particles right. The even split owns almost every arrangement, and the more particles there are, the more total that ownership gets.

Heat flowing from hot to cold, perfume filling a room, milk clouding into coffee: no force drives any of them. Each one is a system sliding into its most numerous macrostate.

03Shuffle a deck

A new deck comes out of the box in order. Here each card is a stripe of colour, so the sorted deck is a smooth gradient. Each riffle cuts the deck roughly in half and lets the halves drop together. Watch the gradient go.

After one riffle you can still see two gradients woven together. Each riffle can at most double the number of runs still in order, so the order fades quickly, though not instantly. In 1992 Dave Bayer and Persi Diaconis showed that around seven riffles is where a 52-card deck becomes properly random. Before that, a sharp card counter can still exploit what's left.

A deck can be ordered in 52! ways, about 8 × 10⁶⁷. Shuffle well and you are almost certainly holding an order no deck in history has ever been in. And try riffling it back to sorted: no law forbids it, yet it never happens. There is one sorted order and a 68-digit number of others. Undoing a mix means landing on one particular arrangement out of all of those.

04Which way is time running?

Three short clips. Each one plays either forward or rewound. Make your call on each.

The pendulum has no friction, so it looks fine in both directions and nobody can call it. You can only tell which way time runs when entropy is being made: ink mixing into water, a ball losing its bounce as heat in the floor. A rewound bounce would need the floor's jiggling molecules to all kick upward together, which is allowed in the same way the letters rebuilding was allowed, and just as unlikely.

The laws of motion don't care which way time runs. Entropy does. The arrow of time you feel, remembering yesterday and never tomorrow, points in the direction entropy grows.

05Entropy is also surprise

In 1948 Claude Shannon needed a way to measure information and ended up with the same formula as Boltzmann. The story goes that John von Neumann told him to call it entropy, since nobody really knows what entropy is, so he'd win every argument.

Shannon's entropy measures how unpredictable something is, in bits. One bit is one fair yes/no question. The more evenly spread the possible outcomes, the more questions you need, on average, to find out which one happened.

Rare outcomes carry the most surprise. Snow in the Sahara would make the news; sun in the Sahara wouldn't. A message you could have predicted tells you nothing, which is also why compression works: a zip file squeezes out the predictable parts and keeps the surprise.

It's the honest way to rate a password, too. A 4-digit PIN has 13.3 bits. Twelve characters picked at random from a full keyboard: about 79 bits. Password1! looks complicated and carries almost none, because it sits near the top of every guessing list.

06What this means for your life

Everything you keep in order sits in a rare, low-entropy macrostate: a tidy kitchen, a clean inbox, a codebase that builds, a friendship kept warm. Left alone, each one drifts toward the messy majority. The room below shows the drift. Every thing has a home on the shelf. Things wander off on their own. Tap one on the floor to put it back.

Turn the habit on with 8 things and the room stays tidy without you. Slide to 30 and the same habit drowns. Trying harder doesn't fix that. A smaller number does.

07The big picture

If everything slides toward higher entropy, how do stars, trees and people exist at all? They are local pockets of order paid for by making more entropy somewhere else.

Earth takes in sunlight as a smaller number of high-energy visible photons and sends the same energy back to space as roughly 20 times as many low-energy infrared ones. Energy in equals energy out, but what leaves carries far more entropy. That gap is the budget plants run on, and plants are the budget you run on. Averaged over a day you burn about 100 watts, a bright old light bulb, and all of it leaves you as heat while you stay organized inside.

Zoom out further. The universe began in an astonishingly low-entropy state, hot and smooth right after the Big Bang. Nobody knows why yet; physicists call the assumption the Past Hypothesis. Everything since (galaxies, stars, life, this page) has happened on the long slope from that start toward a final, featureless maximum that cosmologists call heat death.

We live on the slope. It's the only place anything interesting can happen.

Three things people get wrong

The whole page in five lines

1. Entropy counts the arrangements behind what you see.
2. Systems drift to the macrostate with the most arrangements. That's the second law.
3. Time's arrow points the way entropy grows. Without entropy, rewound and forward look alike.
4. Information is surprise, measured in bits, and it uses the same math.
5. Order is possible anywhere, as long as you pay for it with more entropy somewhere else.

Go deeper

• Veritasium: The Most Misunderstood Concept in Physics the best video on this, and it covers the Sun photons too
• Sean Carroll: From Eternity to Here a whole book on entropy and the arrow of time
• Erwin Schrödinger: What Is Life? (1944) the short book behind "life feeds on order"
• Claude Shannon: A Mathematical Theory of Communication (1948) where information entropy starts
• Landauer's principle why forgetting costs energy