Finite-size correction to a variable-site vacancy model ($⟨n⟩=(N+1)q/(1-q),q=e^{-(ϵ+pv)/(k_{B}T)}$)

If the variable-site model counts every $n$ sector with weight $\left(\genfrac{}{}{0ex}{}{N+n}{n}\right)q^n$, 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 $(N+1)k_{B}T\ln (1-q)$. This correction distinguishes a thermodynamic saddle from an exact finite-system average; changing the microscopic site-creation convention can also change subextensive factors.