= Finite-size correction to a variable-site vacancy model
{title2=$\langle n\rangle=(N+1)q/(1-q),\quad q=e^{-(\epsilon+pv)/(k_BT)}$}
If the variable-site model counts every $n$ sector with weight $\binom{N+n}{n}q^n$, its exact relative partition sum is $(1-q)^{-N-1}$. Differentiating its logarithm gives the displayed exact mean. Minimizing the leading extensive Gibbs free energy instead gives $Nq/(1-q)$, the same result to relative order $1/N$. Its exact relative equilibrium free energy is $(N+1)k_BT\ln(1-q)$. This correction distinguishes a thermodynamic saddle from an exact finite-system average; changing the microscopic site-creation convention can also change subextensive factors.
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