Indistinguishability

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.

i-595dad061fd29981969e7b8f8d615d7e-billiards.png
Distinguishable particles.

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.

More like this

Over the weekend, in an attempt to cheer me up, a kind and generous reader sent me a link [to a *really* wonderful site of crackpot science][adams]. It's a crackpot theory about how physics has it all wrong. You see, there is no such thing as gravity - it's all just pressure. And the earth (and all…
In the comments following the silly accelerator poll, onymous wrote: [T]he point of the LHC isn't to discover the Higgs. No one in their right minds would build a 14 TeV pp collider if their only goal was to discover the Higgs. While it's true that the ultimate goal of the LHC is to discover more…
Part two of our crackpot's babblings are actually more interesting in their way, because they touch on a fascinating mathematical issue, which, unfortunately, Mr. Brookfield is compeletely unable to understand: the Poincare recurrence theorem. Brookfield argues that the second law of…
Remember Granville Sewell? He's the alleged mathematician who wrote the very non-mathematical "A Mathematician's View of Evolution", which I fisked [a few weeks ago](http://scienceblogs.com/goodmath/2006/10/second_law_slop_from_granville…). Well, he's back with a response to the people who…

so this indistinguishable principle actually cuts down on the number of complexities right?

In my opinion the largest threat for California are cataclysms and ecological catastrophes. Not important is how many money we have because one tragedy can us take all.

Matt, I don't believe the quantum-mechanical principle of "identical particles" has anything to do with Gibbs's Paradox - it resolves it, but Gibbs's Paradox can be formulated and resolved even under classical mechanics, and so long as you work with classical mechanics, you'll get the same answers whether your electrons are distinguishable or indistinguishable.

It is only in quantum mechanics that it makes an experimental difference whether electrons are distinguishable or indistinguishable.

It is only in quantum mechanics that we can perform a single simple experimental test that tells us forever and for all time, regardless of the results of further improvements in measuring precision, that two electrons really are identical down to the last decimal place. In classical physics this would just be impossible.