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 a ceiling.
Take the energy of the lowest level as 0 and the ceiling is exactly 0. The horizontal line in the figure is that ceiling, and \mu cannot pass above it.
As the temperature falls, \mu rises. It has no choice, if the number of particles is to stay the same. But the ceiling is there, and sooner or later it runs into it.
Call every level but the lowest, taken together, the excited states. Count how many particles they can hold with \mu pushed right up to the ceiling. That is the most the excited states can hold 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, falling away as the temperature drops.
The horizontal line is the number of particles actually present. Above the temperature where the two cross, the excited states can take all of them. Below that crossing they cannot. The temperature at that crossing is called the transition temperature.
Where do the particles that will not fit go? There is only one place: the lowest level.
The lowest level alone can take 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 strip in the figure shows where the particles are. Above the transition temperature nearly all of them are in excited states. Below it the share held by the lowest level shoots up. What they pile into is 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 leaves 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. By this point an argument that began with counting has reached a cloud of atoms cooled to 10^{-7} kelvin. The tools of statistical mechanics are now all in place.