Cool a gas of bosons without changing the number of particles. The figure shows how the levels fill, and as the temperature drops the particles gather towards the bottom.
So far, nothing surprising. But past a certain temperature something odd happens. Gathering into the lower levels is no longer enough, and a visible fraction of the particles begins to pile into the lowest single level alone.
This is called Bose condensation. It sounds strange, and yet the argument that produces it is nothing but counting. Let us follow it through.
As the previous piece showed, the occupation of a boson level is \dfrac{1}{e^{\beta(\varepsilon - \mu)} - 1}. If \mu rose above the lowest level, the occupation there would come out negative. So \mu has an upper limit.
Take the energy of the lowest level as 0 and the upper limit is exactly 0. The horizontal line in the figure is that limit, and \mu cannot exceed it.
As the temperature falls, \mu rises. It must, if the number of particles is to stay the same. But because of the upper limit, at some temperature \mu reaches it.
Call every level but the lowest, taken together, the excited states. Count how many particles can be in them with \mu raised to the upper limit. That is the largest number of particles the excited states can have at that temperature.
Working it out, that maximum goes as T raised to the power \dfrac{3}{2}. The rising curve in the figure is that, decreasing as the temperature drops.
The horizontal line is the actual number of particles. Above the temperature where the two cross, all of them can be in the excited states. Below that crossing they cannot. The temperature at that crossing is called the transition temperature.
Where do the particles that cannot be in the excited states go? There is only one place: the lowest level.
Only the lowest level can have any number of particles, because as the gap between \varepsilon and \mu approaches 0 the occupation grows without limit. That is why this one level has to be handled separately when counting.
The figure shows which level the particles are in. Above the transition temperature nearly all of them are in excited states. Below it, the fraction of particles in the lowest level rises sharply. They all collect in a single level, so 10^{23} particles come to be in one and the same state.
The figure draws the fraction in the lowest level against temperature. It starts from 0 at the transition temperature and grows as 1 minus \dfrac{T}{T_{\mathrm{c}}} raised to the power \dfrac{3}{2}. At absolute zero it reaches 1, with everything in the lowest level.
This transition differs from an ordinary phase transition in one respect. The particles are not attracting one another. No force between particles has appeared anywhere in the working so far. It happens from the counting alone, from any number being allowed in one level.
It was predicted in 1925 and first made in 1995. An argument that began with counting now explains a cloud of atoms cooled to 10^{-7} kelvin. The basic methods of statistical mechanics are now all in place.