Why time runs one way

A film of a glass shattering looks wrong when played backwards. Yet the law that moves each shard, each molecule of air, runs equally well in either direction. The one-way feel of time is not in the law of motion. It is in counting.

Why it matters

Every equation of motion we have — Newton's, Maxwell's, Schrödinger's — is reversible. Run any of them backwards (for Newton, that means flipping every velocity at some instant) and the system retraces its history exactly. Nothing in the rules says which way is forwards. And yet cream never unmixes from coffee, and heat does not flow from cold to hot on its own. The second law of thermodynamics says the entropy of an isolated system does not decrease. That statement has a direction. The puzzle is where that direction comes from if the rules underneath have none.

The mechanism

Start with two words. A microstate is the complete description: the position and velocity of every particle. A macrostate is what you can actually see — for a gas in a box, say, the fraction of particles in the left half. Many microstates look the same from outside, so each macrostate is a bundle of microstates. Boltzmann's entropy of a macrostate is the logarithm of how many microstates it contains. Bigger bundle, higher entropy.

Now the counting. Put N particles in a box, all in the left half, and take away the wall. Each particle is now either left or right, which gives 2^N ways to sort them. Only one of those ways is "all on the left". Almost all of them — allowing a little slack either side of one half — are "about half and half". For a mole of gas, a few grams, the room the gas gains by spreading is larger, as a ratio, than the known universe is to a single proton. Almost every microstate the box can reach looks like "half and half". The system does not head towards equilibrium because anything pushes it there. It wanders, and nearly everywhere it can wander is equilibrium.

Is it reversible? Yes. Freeze the gas after it has spread, flip every velocity, and it flows back into the left half. Such microstates exist. But they are a vanishing fraction of the spread-out macrostate, and no ordinary process lands on one. Entropy increase is not certain. It is typical, and for large N "typical" is indistinguishable from "always".

This also needs no collisions and no chaos. A gas of particles that never touch one another, each one just bouncing between the walls, spreads out and stays spread, and its Boltzmann entropy climbs to its maximum all the same. What does the work is large numbers plus a starting state that was special.

So the arrow needs one more ingredient: a past that was low in entropy. Given a rare, tidy starting macrostate, the rules push the system towards the vast, common macrostates, and that is the direction we call forwards. Nothing in the dynamics chose it. The initial condition did.

In one breath

The rules of motion have no arrow. Macrostates do, because they are counted: special-looking arrangements are few and ordinary-looking ones are overwhelmingly many. A system that starts in a rare, ordered state almost surely moves into the common ones, and stays there, because there is nowhere else to go. Time's one-way feel is the sound of very large numbers.