Equivalence of Canonical and Microcanonical Ensembles for Euclidean Lattices

Ziyang Zhu

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arXiv · 2607.23526

Equivalence of Canonical and Microcanonical Ensembles for Euclidean Lattices

Ziyang Zhu

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Abstract

We study the equivalence of canonical and microcanonical ensembles for quadratic Hamiltonians on Euclidean lattices. The answer depends on the arithmetic structure of the energy spectrum: exact-energy microcanonical ensembles converge locally to the canonical Gibbs law in the lattice case, but may retain additional arithmetic information in the nonlattice case. We show that this obstruction disappears after replacing an exact energy shell by any fixed-width energy window. Using uniform local central limit theorems, we also obtain sharp asymptotics for the corresponding state counts and recover Bost's thermodynamic entropy.

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Ziyang Zhu. 2026-07-26. Equivalence of Canonical and Microcanonical Ensembles for Euclidean Lattices. https://arxiv.org/abs/2607.23526

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