Entropy additivity from exponential decay of correlations: a coarse-grained operator approach

Thermodynamic extensivity is commonly introduced as a postulate – the homogeneity of degree one in thermodynamic potentials. We provide a constructive derivation of this property from microscopic conditions on the pair potential, without assuming it. Working with the one- and two-particle reduced densities of the $N$-body canonical Gibbs state, we introduce a combined coarse-graining operator $\mathcal{C}$ on single-particle phase space $\mathcal{M}=\mathrm{\Lambda }\times {\mathbb{R}}^3$, producing dimensionless mesoscopic probabilities over spatial–momentum cells $\{V_i\times {\mathrm{\Pi }}_{\alpha }\}$. Under three conditions on the pair potential – stability, temperedness, and exponential cluster decomposition with correlation length $\xi$ – we show, using the Ursell cluster expansion, that the coarse-grained entropy satisfies $$S_{\mathrm{C}\mathrm{G}}=\sum _i{S}_i+O\text{\hspace{0.17em}⁣}\left(\frac{\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}{{\mathrm{\ell }}^d}{e}^{-\mathrm{\ell }\mathrm{/}\xi }\right),$$ where $\mathrm{\ell }\gg \xi$ is the cell diameter. The correction is exponentially suppressed per cell, making entropy additive and recovering the thermodynamic limit of Ruelle and Fisher in explicit operator language. For systems with long-range interactions, where temperedness fails, the correction does not vanish, and non-additivity is quantified through inter-cell mutual information. We further show that spatial averaging does not commute with nonlinear thermodynamic functionals such as the entropy density – a thermodynamic analogue of the cosmological averaging problem – and we derive the generalised Euler relation with explicit surface corrections.