AI 中文总结
研究欧几里得格的正则与微正则系综等价性，通过证明局部能量尺度上的等价性，得到计数渐近性等结果，指出非格情形下精确壳等价性可能不成立，两种情形指数率恢复相关熵函数。
AI 中文摘要
设$\Lambda$为欧几里得格，考虑具有单点能量$H = \lVert\cdot\rVert^2$的$N$个带标签的非相互作用副本。我们证明了在局部能量尺度上正则吉布斯系综和微正则系综的等价性：对于格值能量在可达的精确壳上，否则在固定宽度窗口内。我们得到了每个固定边际的精确计数渐近性和全变差收敛性，但表明在非格情形下精确壳等价性可能不成立。两种情形下的指数率恢复了由博斯特稳定阿雷克罗夫格点计数产生的熵函数。
英文摘要
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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