= Number variance of an interacting defect
{title2=$\operatorname{Var}n=k_BT\,\partial_\mu\bar n$}
With $z=e^{\beta(\mu-\epsilon)}$, $q=e^{-\beta U}$ and $\Xi=1+2z+qz^2$, direct four-state averaging gives
$$
\operatorname{Var}n=\frac{2z(1+2qz+qz^2)}{\Xi^2}=k_BT\,\partial_\mu\bar n.
$$
For $\beta U\gg1$ it has two separated peaks of height $1/4$ at the spin-degeneracy-shifted charge steps. At the lower bare crossing $\mu=\epsilon$, 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.
Back to article page
