Bose-Einstein condensation

Bose-Einstein condensation is the tendency of bosons to collectively and almost exclusively occupy the ground state of a system around and below a specific temperature, known as the condensation temperature or critical temperature. This phenomenon occurs at low temperatures and low energies. Mathematically, it is readily seen from the Bose-Einstein distribution, which gives the average occupation number of each state:

We can see that the distribution has a singularity when the exponential is $1$, which means when $\mu=0$ (and so $z=1$) and $\varepsilon_{i}\to 0$. This leads to a massive spike of occupation numbers around zero energy, which occurs at very low temperatures. This means that at these temperatures, bosons collectively end up in the ground state, leaving other states nearly unoccupied. This is a phase transition.

The asymptotic nature of $\langle n_{i} \rangle$ near the condensation temperature means that it can be seen macroscopically. Also, one- and two-dimensional ideal boson gases do not exhibit this behavior. For a discussion on the exact temperature at which it occurs in an ideal gas of bosons, see Bose gas > Critical condensation temperature.

Details#

The nature of Bose-Einstein condensation lies fundamentally in the antisymmetry of boson wavefunctions. The antisymmetry leads to the $-1$ term seen in the Bose-Einstein distribution above, which as we've seen above causes the singularity at $z=1$. But there's more: the antisymmetry leads to boson systems being well described by Bose functions $g_{k}(z)$. The Bose function of interest is the one that appears in the particle density equation:

Here $\lambda$ is the de Broglie thermal wavelength. For further discussion and origin of this equation, see Ideal gas > In the quantum grand canonical ensemble. We can split this equation into two:

Respectively, these are the ground state and excited state particle numbers, obtained from $N=nV$. When $z$ approaches $1$, the ground state number tends to diverge, leading to the condensation phenomenon. But more can be seen in the excited states. Since $z$ is bounded between $0$ and $1$, so too is $g_{3/2}(z)$ bounded between $0$ and $\zeta(3/2)$, using the Riemann Zeta function. This means that $N_\text{exc}$ is itself bounded:

Importantly, there is an upper limit to the number of excited bosons in a Bose gas. Mind you, not the total number of bosons, specifically just the excited ones. Of course, it is natural that these excited states will host as many bosons as they physically can:

but any more than these and there just isn't any more room. Any excess will be pushed in its entirety down to the ground state, which due to the singularity has effectively unlimited capacity¹. In fact, at a constant volume, when $T\to 0$ and hence $\lambda\to \infty$, this limit tends to $0$. All excited bosons are squeezed out of the excited states back to the ground state.

These two sets of particles, due to their very different behavior, effectively make up two separate phases:

1. A normal phase, consisting of excited bosons distributed as usual.
2. A condensed phase, consisting of a macroscopic amount of bosons all accumulated in the ground state.

At a given volume $V$ and temperature $T$, the onset of the condensation occurs when the number of bosons in the system exceeds the limit for excited bosons:

On the other hand, if we hold $N$ constant instead of $T$, we get

where $T_{C}$ is the critical temperature at which the phase transition occurs.