Film a pair of billiard balls colliding, then play the film backwards. Nobody could tell which version was real. The laws governing the collision run equally well in both directions. Now film milk swirling into coffee and play that backwards. Everybody can tell. The brown liquid separates itself into a white cloud and a black pool, and the cloud jumps back into the jug. It looks like magic, or a fake.
The coffee is nothing but molecules colliding, and each collision is reversible. So where does the one-way direction come from? This puzzle troubled physicists for decades in the 19th century. It is known as the reversibility objection, and Ludwig Boltzmann's answer to it is one of the great ideas of science. Elsewhere in this library, Energy: The One Currency of the Universe and Why Time Is Weirder Than You Think tell you that entropy gives time its direction. This piece shows you why.
A box of particles
Here's the simplest possible version, a model published by the physicists Paul and Tatiana Ehrenfest in 1907. Take a box with an imaginary line down the middle. Start with every particle on the left. Then, again and again, pick one particle at random and move it to the other side. That's all: no force pushes the particles right, and the rule is perfectly even-handed.
Press Run. The particles spread out until the two halves hold roughly equal numbers, and then they jiggle around half-and-half forever. With 4 particles you will often see all four drift back to the left for a moment. With 20, almost never. With 400, the trace becomes a flat line at the middle.
It's just counting
Nothing in the rule prefers the middle. What prefers the middle is counting. "All particles on the left" can happen in exactly one way. "Half on each side" can happen in an enormous number of ways, because it doesn't matter which particles are where. With 100 particles there is 1 arrangement with all of them on the left, and about 1.01 × 1029 arrangements with exactly 50 on each side.
Physicists call a detailed arrangement (which particle is where) a microstate, and a summary you can actually see (how many are on the left) a macrostate. A random shuffle lands you in a typical microstate, and nearly all microstates belong to the mixed-up macrostates. Entropy is a measure of how many microstates a macrostate has. Boltzmann's formula for it, S = k log W, where W is the count, is carved on his gravestone in Vienna's Central Cemetery. (The tidy form of the formula is due to Max Planck, around 1900, building on Boltzmann's work of the 1870s.)
Things don't mix because mixing is pushed. They mix because there are overwhelmingly more ways to be mixed than unmixed.
"Never" is a very large number
This makes the second law of thermodynamics a law of probability, not of iron necessity. Un-mixing is not forbidden. It is just unlikely, in a way that grows with frightening speed:
| Particles | Chance they're all on one chosen side at a given moment |
|---|---|
| 4 | 1 in 16 |
| 10 | 1 in 1,024 |
| 100 | 1 in 1.27 × 1030 |
| 1,000 | 1 in 1.07 × 10301 |
| A litre of air (about 2.5 × 1022 molecules) | 1 in 10 to the power 7.5 × 1021 |
Mark Kac showed that in the Ehrenfest box, if you start with everything on one side, the average wait before that happens again is 2N steps. For just 100 particles, at a billion steps a second, that is about 40 trillion years, roughly 2,900 times the present age of the universe. For a real cup of coffee, the number doesn't fit in any sentence. "Possible but you will never see it" is the honest description.
The real puzzle: why was the past so tidy?
Here's where it gets strange. The counting argument says that from any ordinary state, entropy will almost certainly be higher in the future. But the argument is perfectly symmetric in time. Applied to the past, it says entropy should also almost certainly have been higher in the past. If all you knew was the half-stirred coffee in front of you, the "most likely" history would be a fully mixed cup that spontaneously half-separated a moment ago.
We don't believe that, because we remember pouring the milk. But memory is itself a physical record, made possible by the same one-way flow. So what really breaks the symmetry? The answer most physicists accept is an assumption about the beginning. The universe started in a state of extraordinarily low entropy, and everything since has been the long slide away from it. The philosopher David Albert named this assumption the Past Hypothesis. Roger Penrose estimated how special that starting state was: about 1 chance in 10 to the power 10123. That is a 1 followed by 10123 zeros: there isn't room in the universe to write it out.
Why the universe began like that is not known. It might be explained by a future theory of the Big Bang, or it might simply be how things are. Either way, every arrow you experience follows from it. You remember yesterday and not tomorrow, causes come before effects, eggs break, coffee mixes and you get older, all because the universe started out tidy and has been getting less tidy ever since.
A word on "disorder"
Entropy is often described as disorder, and that's a useful shorthand, but it can mislead. Entropy counts the microscopic arrangements consistent with what you can see. Sometimes a process makes something look more orderly: water freezing into neat ice crystals, for example. But freezing gives off heat, and that heat raises the entropy of the surroundings by more than the crystals lowered it. The rule is about counting everything, not about how tidy one part looks.
The laws of molecular collisions don't care about the direction of time, but counting does. Mixed states can happen in astronomically more ways than unmixed ones, so a system wandering at random ends up mixed and stays there. That is the second law: a law of overwhelming probability, measured by Boltzmann's S = k log W. The same counting would predict a messier past too. Time's arrow exists because the universe began in an extraordinarily low-entropy state, and why it did remains an open question.
Try this: set the box to 4 particles and count how often they all gather on one side in a minute of running. Then try 10 and 20. You are watching the second law emerge from nothing but chance.
Sources. The Ehrenfest urn model (1907), the reversibility (Loschmidt) and recurrence (Zermelo) objections it addressed, and Kac's results that entropy rises steadily from an unmixed start and that the mean recurrence time from an all-on-one-side start is 2N steps: Wikipedia, "Ehrenfest model", and the literature it cites. Boltzmann's gravestone and Planck's form of S = k log W: Atlas Obscura and history-of-physics sources. The Past Hypothesis: D. Albert, Time and Chance (2000). Penrose's 1 in 1010123: R. Penrose, The Emperor's New Mind (1989). Computed in Python and node: C(100,50) = 1.009 × 1029; 2100 = 1.27 × 1030; 21000 = 1.07 × 10301; molecules per litre at 20 °C and 1 atm from the ideal gas law = 2.50 × 1022; 2100 steps ÷ 109 per second = 4.0 × 1013 years = 2,900 × 13.8 billion years. The interactive is the Ehrenfest model exactly; its arrangement counts are computed exactly with big integers.
- 📘 From Eternity to Here — Sean Carroll (the arrow of time and the Past Hypothesis, for general readers)
- 📘 The Order of Time — Carlo Rovelli (short, poetic and rigorous)
- 🌐 The Ehrenfest model (the maths behind the box)
- 🌐 Atlas Obscura: Boltzmann's grave