Bose-Einstein Condensates, pt. 4

Well, we've explored some groundwork in three previous posts and so it's time to put it all together. Why exactly do bosons have such weird behavior at very low temperatures, with a large fraction of their number crowding into a single quantum state? Let's plow on. If you're not familiar with the mathematics or the physics, don't worry. What you've absorbed from the previous posts will be fine - you don't need to know the details to understand the big picture.

The number of bosons is given by the grand partition function Q in the following way:

Each term in the sum is the expected number of particles in that particular energy state. We can't do this sum by hand, so we'd like to replace it by an integral. We can do this by multiplying the whole thing by the density of states. However, our density of states is an approximation. A very good one in most cases, but in the case of low-temperature bosons we run into a problem. The density of states is zero for a state with zero energy. And while there's no such thing as a truly zero-energy state, the actual ground state is close enough so that the density of states fails to take the ground state into account in any reasonable way. To work around this, we'll keep the zero state outside the integral:

Now that integral is the function we talked about last Tuesday. It takes a maximum value where chemical potential z = 1. The important point is that the integral is finite. For a given beta (proportional to the inverse of the temperature) that integral will take a particular number as its value, and that number will be the number of particles in all the excited states. If the number of particles in the system goes over that value, the entire number of particles greater than that number have to be in the N₀ state. There's no choice, there's no room in the excited states.

Now if you actually do the integral - the details of which I'll skip because they're pretty unenlightening - you can calculate that the maximum number of particles in all of the excited states combined is:

Which decreases as T decreases. Eventually it's going to be smaller than the number of particles in the system, and there's no choice but for the the excess to be pushed into the ground state. Quantum magic.

For clarity one can rewrite this in terms of the particles/volume n and solve for the critical temperature below which condensation sets in:

Now we've made all kinds of approximations and it will turn out that in fact our description is qualitatively right but quantitatively rather off-base. Still, it was pretty much tis argument that first predicted the phenomenon, and it's refinements of this model that very accurately describe it today.