Grand-potential particle-number variance ($\mathrm{Var}N=-k_{B}T{\partial }_{\mu }^2\Phi$)

Differentiating the grand partition function at fixed temperature and volume gives $⟨N⟩=-{\partial }_{\mu }\Phi$ and $\mathrm{Var}N=-k_{B}T{\partial }_{\mu }^2\Phi =k_{B}T{\partial }_{\mu }⟨N⟩$. It follows because derivatives of the logarithm of the state sum generate cumulants of particle number. Nonnegative variance makes the grand potential concave in chemical potential in a stable grand-canonical description.