Number variance of an interacting defect ($\mathrm{Var}n=k_{B}T{\partial }_{\mu }\bar{n}$)

With $z=e^{\beta (\mu -ϵ)}$, $q=e^{-\beta U}$ and $\Xi =1+2z+q{z}^2$, direct four-state averaging givesFor $\beta U\gg 1$ it has two separated peaks of height $1/4$ at the spin-degeneracy-shifted charge steps. At the lower bare crossing $\mu =ϵ$, three microstates share equal limiting weight, giving $\bar{n}=2/3$ and variance $2/9$. The upper bare crossing similarly gives mean $4/3$ and variance $2/9$. Between crossings occupation is pinned near one and fluctuations are suppressed. The identity is grand-canonical; fixing total particle number would change the fluctuation question.