AI 中文总结
本研究基于平稳性与无偏性公设，建立适用于任意大小经典无相互作用多体系统的普适系综理论，推导各统计系综下熵的精确表达式，且其渐近形式与大偏差理论预测一致。
AI 中文摘要
在热力学极限下，多体系统的平衡态可由三对共轭热力学变量$E/T$、$V/P$、$N/\boldsymbol{\beta}$表征。在此极限下，所有系综的热力学性质在$E$、$V$、$N$的主导阶上等价。然而对于有限大小的系统，这种系综等价性不再精确成立，不同系综间的热力学性质可能存在显著差异。为严格量化这些有限尺寸效应，亟需建立适用于任意大小系统的普适系综理论，提供熵的精确表达式，进而得到所有平衡热力学量的精确值。本研究基于两个公设提出一种理论，确定了任意大小经典无相互作用多体系统在所有统计系综下的熵的精确表达式：一是**平稳性**，要求物理定律在时间平移下不变；二是**无偏性**，要求平衡混合态在给定约束下最大化熵。此外，研究表明不同系综下得到的熵表达式收敛到共同的渐近形式$S \backsim \boldsymbol{\beta}\text{ln}\bigg(\bigg(\frac{4 \boldsymbol{\beta} m e E}{3N}\bigg)^{3N/2} \boldsymbol{\beta} \bigg(\frac{V}{N}\bigg)^N \bigg)+N$，与大偏差理论的预测一致。
英文摘要
In the thermodynamic limit, the equilibrium state of a many-body system can be characterized by three pairs of conjugate thermodynamic variables: $E/T,V/P,N/μ$. In this limit, the thermodynamic properties in all ensembles are equivalent up to the leading order of $E,V,N$. However, for systems of finite size, this ensemble equivalence is no longer exact, and the thermodynamic properties may differ substantially among ensembles. To quantify these finite-size effects rigorously, it is desirable to develop a universal ensemble theory applicable to systems of arbitrary size, providing exact expressions for entropy and, thereby, giving rise to the precise value of all equilibrium thermodynamic quantities. In this work, we propose a theory that determines the exact entropy expressions for classical non-interacting many-body systems of arbitrary size across all statistical ensembles, based on only two postulates: \textbf{stationarity}, requiring that the physical laws be invariant under time translation, and \textbf{unbiasedness}, requiring that the equilibrium mixed state maximize the entropy subject to the prescribed constraints. Moreover, we show that the entropy expressions obtained in different ensembles converge to the common asymptotic form $S \asymp \ln\!\left( \left( \frac{4πm e E}{3N} \right)^{3N/2} \cdot \left(\frac{V}{N}\right)^N \right)+N$, consistent with the predictions of the large deviation theory.
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