2.2.1.3 Distribution of Bosons

In the case of symmetric wave functions the occupation numbers of quantum states are not limited and can take on any values. The series (2.27) converges only if $\exp[(\mu-\epsilon_{k})/k_{B}T]<1$. This condition is valid for any $\epsilon_{k}$ including zero. Thus the chemical potential must be negative while for fermions it can take both positive and negative values.

The series (2.27) represents a geometric progression and its value can easily be obtained:

$$\Omega_{k}=k_{B}T\ln\biggl(1-\exp\biggl(\frac{\mu-\epsilon_{k}}{k_{B}T}\biggr)\biggr).$$ (2.33)

Using (2.29) one obtains from (2.33) for the average number of bosons:

$$\langle n_{k} \rangle = \frac{1}{\exp\bigl(\frac{\epsilon_{k}-\mu}{k_{B}T}\bigr)-1}.$$ (2.34)

Considering the equilibrium phonon gas the number of phonons in the phase space element $d\vec{r}d\vec{k}$ can now be written as^2.9:

$$dN_{ph}=\frac{1}{\exp\bigl(\frac{\hbar\omega(\vec{k})}{k_{B}T}\bigr)-1}\frac{d^{3}kd^{3}r}{(2\pi)^{3}},$$ (2.35)

where $\omega$ is the phonon frequency. S. Smirnov: