Continuing our “basic concepts with no math” series, here’s one of the most important for quantum mechanics and statistical mechanics.
To start, imagine a billiards table. It has an assortment of billiard balls, each numbered and colored differently. You can imagine randomly knocking them around the table, sort of in analogy to gas molecules bouncing around a room. As we’ve discussed, it’s very likely that on average the balls will be distributed more or less evenly around the table. You will rarely just happen to end up with all of them on one side of the table. By pure chance it might happen, but for a larger and larger number of balls it gets less and less probable.
In any event, think about a situaiton where all but one of the billiard balls are on one side of the table. You might have the 8-ball on one side and the rest on the other. Or it might be the 3-ball on one side and the rest on the other. Or any of the various other possibilities. In every case you’re still in the overall “all but one on one side” state, but there’s a number of different ways to be in that state. And you can tell those ways apart.
Could we do this for, say, a gas of electrons banging around in an enclosure? Mathematically we can, and a page or so into the algebra of the thermodynamics we’ll recognize a problem called the Gibbs paradox. It’s going to turn out that mathematically the entropy can then be made to behave in impossible ways. And clearly entropy is such a fundamental concept that something’s got to be wrong in our math.
And there is. It turns out that you can’t tell electrons apart. They’re not labeled the “8-ball electron” and so forth. And they can’t be labeled even in theory. Every electron is as far as we know exactly identical to every other one. And that has profound consequences. There’s no longer nearly so many ways to achieve the “one electron in the right half of the container, the rest in the left” state, because it doesn’t matter which electron is on the right. The universe can’t tell them apart. When we go back and adjust our equations to take this into account, we find that now the mathematics works and the equations describe what we actually observe experimentally.
That’s just one aspect of the thermodynamics of identical particles. It has interesting and subtle consequences in quantum mechanics as well. That’s another good story, which deserves its own look later.