Perturbation Theory of the Free Energy via the Mesoscopic Combined Partition Function

We develop a systematic perturbation theory for the Helmholtz free energy of a classical $N$-body system within the mesoscopic framework of~\cite{OsanoMeso,OsanoExtensivity}. The combined coarse-graining operator $\mathcal{C}={\mathcal{C}}_x\circ {\mathcal{C}}_p$ acting on single-particle phase space partitions it into product cells $C_{i,\alpha }=V_i\times {\mathrm{\Pi }}_{\alpha }$ and generates a mesoscopic partition function ${\mathcal{Z}}_{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{o}}(\lambda )$ whose reference level factorises by the multinomial theorem: ${\mathcal{Z}}_{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{o}}^{(0)}=(Z_1^{(0)}{)}^N$. Perturbation theory for ${\mathcal{F}}_{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{o}}(\lambda )=-k_{B}T\ln {\mathcal{Z}}_{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{o}}(\lambda )$ in the inter-cell perturbation ${\mathcal{V}}_{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{o}}$ yields the mesoscopic Gibbs–Bogoliubov inequality and an exact coupling-parameter integration formula. The full free energy satisfies \begin{equation*} F(λ)=$\mathcal{F}${\rm meso}(λ)-k_BT!\sum{i<j}I(i,j;λ)+O!\left(|Λ|\ell^{-d}e^{-2\ell/ξ}\right), \end{equation*} where the inter-cell mutual informations $I(i,j;\lambda )$ are the corrections identified in the extensivity analysis. The first-order theory recovers the van der Waals equation and the Barker–Henderson result; the second-order term converges to the structure-factor formula in the fine-cell limit. For long-range interactions, factorisation fails, and the mutual-information corrections quantify the resulting non-extensivity.