Perturbation Theory of the Free Energy via the Mesoscopic Combined Partition Function
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866913140637171712 |
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| author | Osano, Bob |
| author_facet | Osano, Bob |
| contents | 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 $C_{i,α}=V_i\timesΠ_α$ and generates a mesoscopic partition function $\mathcal{Z}_{\rm meso}(λ)$ whose reference level factorises by the multinomial theorem: $\mathcal{Z}_{\rm meso}^{(0)}=(Z_1^{(0)})^N$. Perturbation theory for $\mathcal{F}_{\rm meso}(λ)=-k_BT\ln\mathcal{Z}_{\rm meso}(λ)$ in the inter-cell perturbation $\mathcal{V}_{\rm meso}$ 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 $I(i,j;λ)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18121 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Perturbation Theory of the Free Energy via the Mesoscopic Combined Partition Function Osano, Bob Statistical Mechanics 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 $C_{i,α}=V_i\timesΠ_α$ and generates a mesoscopic partition function $\mathcal{Z}_{\rm meso}(λ)$ whose reference level factorises by the multinomial theorem: $\mathcal{Z}_{\rm meso}^{(0)}=(Z_1^{(0)})^N$. Perturbation theory for $\mathcal{F}_{\rm meso}(λ)=-k_BT\ln\mathcal{Z}_{\rm meso}(λ)$ in the inter-cell perturbation $\mathcal{V}_{\rm meso}$ 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 $I(i,j;λ)$ 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. |
| title | Perturbation Theory of the Free Energy via the Mesoscopic Combined Partition Function |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2605.18121 |