= Grand-potential particle-number variance
{title2=$\operatorname{Var}N=-k_BT\,\partial_\mu^2\Phi$}
Differentiating the grand <partition function> at fixed <temperature> and <volume> gives $\langle N\rangle=-\partial_\mu\Phi$ and $\operatorname{Var}N=-k_BT\partial_\mu^2\Phi=k_BT\partial_\mu\langle N\rangle$. 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.
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