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

TL;DR

This paper derives thermodynamic extensivity from microscopic conditions using a coarse-grained operator approach, showing entropy additivity under certain conditions and quantifying non-additivity for long-range interactions.

Contribution

It provides a constructive derivation of entropy additivity from microscopic pair potential conditions without assuming thermodynamic homogeneity.

Findings

Entropy becomes additive with exponentially suppressed corrections under stability, temperedness, and exponential clustering.

For long-range interactions, non-additivity is quantified via inter-cell mutual information.

Spatial averaging does not commute with nonlinear thermodynamic functionals, affecting entropy density calculations.

Abstract

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 $\mathcal{M}=\Lambda\times\mathbb{R}^3$, producing dimensionless mesoscopic probabilities over spatial--momentum cells $\{V_i\times\Pi_\alpha\}$. 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 \[S_{\mathrm{CG}}=\sum_i S_i+O\!\left(\frac{|\Lambda|}{\ell^d}e^{-\ell/\xi}\right),\] where $\ell\gg\xi$ 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.