Imagine that there exists a third type of particle, which can share a single-particle state with one other particle of the same type but no more. Thus the number of these particles in any state can be $0,1$ or $2$ . Derive the distribution function for the average occupancy of a state by particles of this type, and plot the occupancy as a function of the state's energy, for several different temperatures.

We are given a particle that can share a single-particle state with one other particle of the same type but no more and the number of these particles in any state can be $0,1$or $2$.

The grand partition function or Gibbs sum is,

$Z=\sum _n{e}^{\frac{-n(ϵ-\mu )}{kT}}$

According to the given problem, the number of particles of third type in any state can be $0,1$or $2$. That is n can be $0,1\text{or}2$.

$$\begin{gathered} Z=\sum _{n=0}^2\exp \left[\frac{-n(\epsilon -\mu )}{kT}\right] \\ Z=\exp \left(0\right)+\exp \left[\frac{-(\epsilon -\mu )}{kT}\right]+\exp \left[\frac{-2(\epsilon -\mu )}{kT}\right] \\ Z=1+\exp \left[\frac{-(\epsilon -\mu )}{kT}\right]+\exp \left[\frac{-2(\epsilon -\mu )}{kT}\right] \end{gathered}$$

Let $x=\exp \left[\frac{-(\epsilon -\mu )}{kT}\right]$,

$Z=1+x+x^2$

The probability of state being occupied by n-particles is,

$$\begin{gathered} \mathrm{P}\left(\mathrm{n}\right)=\frac{1}{Z}\exp \left(\frac{-n(\epsilon -\mu )}{kT}\right) \\ =\frac{1}{Z}{x}^n \\ \mathrm{P}\left(n\right)=\frac{x^n}{1+x+x^2} \end{gathered}$$

The average occupancy of state by particles of this type is,

$$\begin{gathered} \overline{n}=\sum _{n=0}^{2}n\mathrm{P}\left(n\right) \\ =0·P\left(0\right)+1·P\left(1\right)+2·P\left(2\right) \\ =\frac{1·x^1}{1+x+x^2}+\frac{2·x^2}{1+x+x^2} \\ \overline{n}=\frac{x+2x^2}{1+x+x^2} \end{gathered}$$

So, the distribution function for the number of these particles in any state can be $0,1$or $2$is,

$$\begin{gathered} n=\frac{x+2x^2}{1+x+x^2} \\ x=\exp \left(\frac{-(\epsilon -\mu )}{kT}\right) \end{gathered}$$

The graph is as follows,