Why Your Coffee Never Un-Stirs

Pour milk into black coffee and give it a stir. White curls twist through the dark, stretch into ribbons, and fade into one even tan.

Now play the clip backwards. The tan separates. The ribbons knot themselves into curls, the curls pull together into a clean white stream, and the stream climbs back up into the jug. You know it's fake before you can say why.

Here's what makes that strange. Film two billiard balls colliding and play it backwards, and it looks perfectly fine. Nobody can tell which way the film is running. The milk is just molecules colliding with molecules, and every one of those collisions is as reversible as the billiard balls'. Cut the milk clip into single collisions and each one runs backwards without breaking a single rule. Put them back together and the whole clip is impossible.

So where does the one-way street come from?

The short answer

Here's the whole answer in five steps. The rest of this post takes them one at a time and shows how we know each one.

1. Cost points nowhere. Every difference between two things has a price, and the price is the same both ways. Doubling costs exactly 1/4. Halving costs exactly 1/4. Cost can't tell the past from the future.
2. The balance comes back. The record doesn't. Post an entry and then its opposite, and the books balance exactly as before, with two more lines in the ledger. The ledger only adds. Time's direction is the direction the record grows.
3. A perfect swap makes no entropy. The entropy a step produces is the information it throws away. A step with an exact opposite partner can always be undone, so it throws nothing away.
4. A merge makes entropy. A step that merges different arrangements into one throws the detail between them away, and the entropy it produces is exactly that detail. No merge, no entropy.
5. The count. One piece of the ledger, counted exactly under its four rules: its starting state carries 6.25 bits of information about where it began, its first tick throws away 35.7% of that, and its balance comes back on average every 76 ticks. Its record never comes back.

Why this matters: the arrow of time stops being a mystery stapled onto physics. It has a place you can point to: the ledger only takes new entries. And entropy stops being a vague word for disorder. It's a number you can count, the detail a step threw away. Your coffee forgets how it was stirred. The ledger never forgets anything.

Previously

If you missed Parts 1 and 2, here's what you need, in about a minute.¹

Reality is a permanent record. Every event is written into it, and nothing is ever erased. We call it the ledger. If it had a personality, it would be unflappable, extremely thorough, and armed with one response to everything that has ever happened: Noted. The past still exists in it, line by line, in pen.

Part 2 left one loose end. It said eggs break and don't unbreak "for the usual reason, heat and disorder," and then moved on to "a second, quieter arrow built into the ledger itself." This post takes that usual reason seriously: what "disorder" actually is, and why it only goes one way.

Here's today's route. Same deal as before: we'll stop after each step to zoom out.

Step 1: Cost points nowhere

The price of a difference is the same in both directions. Here's what that means.

Reality charges for differences. Whenever two things are compared, the comparison has a price, and the price depends on how lopsided it is. Write the lopsidedness as a ratio, x. If x = 1, the two things match. If x = 2, one is double the other. If x = 1/2, one is half the other. The price is

J(x) = (x + 1/x) ÷ 2 − 1

It isn't a formula picked to look neat. Given how comparisons have to combine, it's the only price that works.² Put some numbers in:

A perfect match is free. Anything lopsided costs something, and the more lopsided, the more it costs.

Now read down the table. Doubling costs 1/4. Halving costs 1/4. Quadrupling and quartering both cost 9/8. Swap x for 1/x and the formula can't tell the difference, because x + 1/x comes out the same either way.

So going backwards isn't more expensive than going forwards. There's no surge pricing for time reversal. Take the reversed milk clip and list its steps: each one, run backwards, costs exactly what it cost going forwards. The price list has no objection to the reversed clip.

That clears the first suspect. Whatever makes time one-way, it isn't cost.

Zoom out. Step 1 is done. Every difference has a price, and the price is the same in both directions: J(2) = J(1/2) = 1/4. Cost can't tell the past from the future. Next, and why: if cost doesn't point, something else does. The next place to look is the ledger's bookkeeping.

Step 2: The balance comes back. The record doesn't.

Post an entry and then its opposite, and the books come back exactly. The record doesn't. Open the ledger and look at what an entry is.

There are three accounts, one for each dimension of space. Each account has two columns, debits and credits. An entry adds one to one column of one account: a debit in account 2, say, or a credit in account 3. Three accounts and two columns make six possible entries.

An account is balanced when its debits and credits match. In the drawings, each account is a dial whose needle shows debits minus credits, so a balanced account reads 0. The dials are a drawing of the accounts. The accounts are the real thing. Next to the dials sits the record: one line for every entry ever posted.

Start with all three dials at 0 and a blank record.

Post a debit in account 1. The needle moves to +1. The record has one line.

Post a credit in account 1. The needle moves back to 0. The record has two lines.

The books are balanced again, exactly as before. The record isn't back where it started, and it can't be, because the ledger has no entry that takes a line away. A debit adds one to a debit column. A credit adds one to a credit column. Nothing subtracts. The reverse of an entry would have to subtract. By Step 1 it would cost the same as the entry. It still isn't on the menu.

Going backwards isn't expensive. It isn't an entry. That's all "the ledger has no eraser" means in the accounts.

What the ledger does allow is the opposite entry. Post it, and the books retrace their steps. That's all a collision run backwards is: the same motion, written down as new lines.

So the same entries do two different things, depending on what you watch. Watch the balance, and it goes round in circles. Watch the record, and it runs in a straight line, one more line at a time, and never comes back.

Time's direction is the direction the record grows.

It follows from what an entry is, and you can count it. Write eight entries in a row, each one of the six possible. That gives 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 = 1,679,616 possible histories. In 44,730 of them, the books end up balanced: the balance came home. In all 1,679,616 of them, the record is exactly eight lines long. Not one history ends with a record it had before.

Zoom out. Step 2 is done. The balance can come back (in 44,730 of the 1,679,616 eight-entry histories it does) and the record never does, because an entry only adds. That's where the direction lives. Next, and why: a balance coming back doesn't mean nothing was lost. To see what a coffee cup loses, we need two kinds of steps: the kind that lose nothing, and the kind that lose something.

Step 3: A perfect swap makes no entropy

A step that can be undone exactly throws nothing away, so it produces no entropy. To see it at work, we need a case small enough to count.

Meet Gary again, the drummer from Part 1 who flips a coin before every beat. This time Gary flips three coins. There are 2 × 2 × 2 = 8 possible sequences, and each one is an arrangement of the coins:

HHH, HHT, HTH, THH, HTT, THT, TTH, TTT

Call the chances you give each arrangement the picture. Two pictures matter.

• The settled picture. You haven't looked, and the coins were shuffled fairly. Every arrangement gets 1 chance in 8.
• The sharp picture. You've looked. Gary flipped HHT. That arrangement gets 1 chance in 1, and the other seven get none.

The sharp picture knows more than the settled one. How much? Count the yes/no questions. Pinning down one arrangement out of 8 takes three: is the first coin heads, is the second, is the third? One yes/no question's worth of information is a bit. So the sharp picture carries 3 bits that the settled picture doesn't. (Another way to write it: log₂ 8 = 3, because 2 × 2 × 2 = 8.)

Now give the coins a swap: turn every coin over. HHT becomes TTH. Every arrangement has exactly one partner this way, and turning every coin over again brings you home.

What did the swap throw away? Nothing. From TTH you can read off HHT. The picture after the swap is still sharp, still 3 bits. The entropy it produced is 0. Swap a thousand times, or a million, and it's still 0. Gary is still not getting the gig.

The accounts have an exact swap like this too. It acts on the dials, not on the record, so it isn't an entry. Trade every debit for a credit and every credit for a debit, and each dial flips its sign: a needle at +1 goes to −1. In Step 1's language, a needle at +1 and a needle at −1 are mirror partners, like x and 1/x: a doubling and a halving. Mirror partners cost the same and sit equally far from balance, so the settled picture gives them the same chance, and the swap shuffles the arrangements without changing any arrangement's settled chance. Do it twice and every dial is home.

How do we know? A swap only renames the arrangements, and it gives each one a partner with the same settled chance. Renaming like that can't change how much a picture knows.

Zoom out. Step 3 is done. A step with an exact opposite partner throws nothing away, so it produces no entropy, however many times you repeat it. Next, and why: if swaps make none, entropy has to come from steps that have no exact partner. The clearest of those is the merge.

Step 4: A merge makes entropy

A merge throws detail away, and the entropy it produces is exactly that detail. Here's the same case with a different step.

A merge keeps a short summary of the arrangement and throws the rest away. Call the summary a report, so that "the record" always means the ledger. Say Gary's report is just how many heads: two. The report doesn't say which two coins. The merge then redraws the rest the way a fair shuffle would: among the arrangements that match the report, every one gets the same chance.

Which arrangements match "two heads"? HHT, HTH and THH. After the merge, each of those three has 1 chance in 3. The other five have none.

How much does this picture know beyond the settled one? For each arrangement still possible, compare its chance with its settled chance: 1/3 against 1/8, a ratio of 8/3. The information is log₂(8/3) = 1.415 bits. (In general, add up, over the arrangements still possible, each one's chance × log₂(its chance ÷ its settled chance). Here that's three terms of the same size.)

The picture went from 3 bits to 1.415. The merge threw away 3 − 1.415 = 1.585 bits. That number is log₂ 3. It's exactly the answer to the question "which of the three?", the detail the report doesn't keep. That's the entropy this step produced.

Change what the report keeps, and the amount changes:

(Starting from HHT. Here the bits thrown away are log₂ of how many arrangements look the same.)

Read the table from the top. A report that keeps every distinction throws nothing away, which is Step 3 again: no merge, no entropy. A blurrier report never throws away less. Throw everything away, and the picture lands on the settled one with all 3 bits gone.

How do we know this isn't just a fact about coins? It's exact for any step that keeps a report and redraws the rest the way the settled picture would, and it holds just the same when the settled chances aren't equal: you weight by them. Information thrown away can't be negative, so a step like this never produces negative entropy. That's the second law of thermodynamics for steps of this kind. If you've met S = k log W in a physics class, W is the number of arrangements the report can't tell apart. Here it's 3.

Now the coffee. Draw the cup like Gary's coins. Right after the pour, the picture is sharp: it knows the curl of milk, and it knows the fine detail underneath, which molecule is where. Each collision on its own runs fine backwards. But the cup keeps only the coarse report (how much milk is in each part of the cup, how warm it is), and the fine detail underneath gets merged away. Merge after merge, the picture slides toward the settled one: an even tan. The detail that made the curl isn't in the cup anymore. Un-stirring needs it back, and the cup doesn't have it. From "two heads", you can't tell whether it was HHT, HTH or THH.

But it isn't gone from reality. Every collision that stirred the cup is an entry in the ledger. The cup forgets. The ledger doesn't.

So there are two one-way streets. The ledger only adds. A merge can't be undone from its report.

Zoom out. Step 4 is done. A merge produces exactly the entropy of the detail it throws away, and a step with no merge produces none. Next, and why: Gary's coins are the smallest case of the accounting. To see it in a real law, we take one piece of the ledger and count.

Step 5: The count

One piece of the ledger has exact numbers: 6.25 bits, 35.7%, 76 ticks. This is where the arrow stops being a slogan and becomes numbers. Here's where they come from.

Take one piece of the ledger: the three accounts from Step 2 and a law for what it does each tick. (This is a different view of the books from the two clocks of Parts 1 and 2. Here we watch only three accounts, and on some ticks nothing posts to them.) Each tick, the piece does one of two things. It holds: nothing is posted and the dials stay put. Or it posts one entry, one of the six.

Its three dial readings, like (+1, 0, 0), are its arrangement. Balanced is (0, 0, 0). One entry moves one dial by one notch, so arrangements one entry apart are neighbors, and the steps from balance are the total distance the dials sit from 0: (+1, +1, 0) is two steps away. Dial readings aren't a place in space. They're account balances.

Four rules pin down the piece's law:

1. What happens next depends only on how things stand now.
2. Each tick the piece holds, or posts one entry.
3. Once everything has settled, every crossing from one arrangement to a neighbor is matched by as many crossings back, and an arrangement n steps from balance is φⁿ times less likely than balance. Here φ = 1.618..., the golden ratio from Part 2.
4. The chance of holding at one arrangement times the chance of holding at its neighbor equals the chance of stepping across times the chance of stepping straight back.

Exactly one law meets all four, so nothing is left to tune.³ At balance, that law has the piece hold with chance 0.184 and post each of its six possible entries with chance 0.136. (0.184 + 6 × 0.136 = 1.) Nobody picks those two numbers. They fall out of the four rules, and every number from here on is exact arithmetic on that law.

The settled picture. Rule 3 gives the settled chances a shape we can add up. In one account, balance has weight 1, one step either way has weight 1/φ each, two steps either way 1/φ² each, and so on. Add it all up: 1 + 2 × (1/φ + 1/φ² + 1/φ³ + ...). The bracket is 0.618 + 0.382 + 0.236 + 0.146 + ..., which adds up to 1.618, φ itself. So one account's total is 1 + 2φ = 4.236 = φ³. Three accounts multiply: φ³ × φ³ × φ³ = φ⁹ = 38 + 17√5 = 76.01. (The 9 is three accounts times 3 apiece.) So balance has a settled chance of exactly 1 in 76.01. It's the single likeliest arrangement, and it's still rare.

The information. Start the piece at balance on a blank page. That's a sharp picture: balanced for sure, against a settled 1 in 76.01. Its information is log₂ 76.01 = 6.25 bits. That's how much the picture knows about where the piece started: about six and a quarter yes/no questions' worth.⁴

The first tick. After one tick, the picture is: balanced with chance 0.184, and each of the six neighbors with chance 0.136. The settled chances are 1/76.01 = 0.01316 for balance and 0.01316 ÷ φ = 0.00813 for each neighbor. Same formula as Step 4:

From 6.25 down to 4.02: the first tick throws away 2.23 bits, which is 35.7% of everything the start knew. Every tick after that throws away some of what's left, and the total over all the ticks is exactly the 6.25 bits you started with. The information falls every tick and never refills.

The return. Meanwhile the piece's balance keeps coming back. It's certain to, and on average it takes φ⁹ = 76.01 ticks. That's the same 76 as before, and it's no coincidence. The more special an arrangement is, the longer you wait on average to be back in it, and the information it carries is the log of that wait. 6.25 bits and 76 ticks are one number, written two ways: log₂ 76 = 6.25.

The record never returns. After the piece's first entry, the record is never blank again. The balance comes back every 76 ticks on average. The record never comes back at all.

Two clocks run in the piece. One comes home again and again. The other only goes forward. (Once the piece has settled, a film of its balance looks the same run backwards, like the billiard balls. The direction is in the settling and in the record.)

Here's where Yeah-But clears his throat.

"Yeah, but if the balance comes back every 76 ticks, why doesn't my coffee un-stir?"

Fair question, and the numbers answer it. One piece comes back every 76 ticks on average. Take two pieces that don't talk to each other. For both to be balanced at the same moment, you wait for a 1-in-76 event twice over: about 5,778 ticks. Ten pieces: about 6.4 billion billion ticks. A hundred pieces: a wait with 189 digits. Every extra piece adds almost two more. A cup of coffee is made of far more than a hundred pieces, and no arrangement of them is easier to return to than balance. Coming back isn't forbidden. The wait is just absurd.

And if it ever did happen, the ledger would be longer by every entry along the way. The balance can come home. The record can't.

Zoom out. Step 5 is done. One piece of the ledger, counted exactly: 6.25 bits at the start, 35.7% of them gone in the first tick, balance back every 76 ticks on average, and a record that never comes back. Next, and why: the honest edge of all this, what's still being worked out, and then the whole trip in one breath.

What's still being worked out

Three things aren't finished: the proof that every step in a real cup of coffee follows Step 4's accounting, which is the second law of thermodynamics in full; why the piece must follow its last two rules; and how this arrow fits with the one in Part 2's clock table.

The whole trip, in one breath

Cost points nowhere: doubling and halving both cost 1/4. The balance comes back and the record doesn't, because the ledger only adds, and time's direction is the direction the record grows. A perfect swap throws nothing away and makes no entropy. A merge throws detail away and makes exactly that much. For one piece of the ledger, the start carries 6.25 bits, the first tick throws away 35.7% of them, the balance comes back every 76 ticks on average, and the record never comes back. Reality keeps two clocks, one that comes home again and again and one that only goes forward. Your coffee forgets how it was stirred. The ledger doesn't. That's the one-way street, and it's why yesterday is still there.

(The Ledger would like you to know that your coffee has gone cold while you read this. It noted that. It can't un-note it.)