Bose–Einstein statistics

Remarks

The grand partition function of one state is the geometric series $\sum_{n \ge 0} e^{-n\beta(\varepsilon - \mu)} = 1/(1 - e^{-\beta(\varepsilon - \mu)})$. It converges exactly when $\varepsilon > \mu$, which is why $\mu$ must lie below every energy level. Taking the mean of $n$ with these weights gives the formula above.

Example: for photons $\mu = 0$, and a mode of energy $\varepsilon = k_B T \ln 2$ has $\bar n = 1/(2 - 1) = 1$.

Non-example: if $\mu > \varepsilon$ for some level, the series diverges and no equilibrium state exists, so the hypothesis $\mu < \varepsilon$ cannot be dropped. When $\mu$ approaches the lowest level, the occupation of that level grows without bound; this is the mechanism behind Bose–Einstein condensation.

When $\beta(\varepsilon - \mu) \gg 1$, $\bar n$ is close to the Maxwell–Boltzmann value $e^{-\beta(\varepsilon - \mu)}$.