AI 中文总结
该研究针对统计力学中构形配分函数的逆问题，证明单阱势下简谐振子是唯一与高斯构形配分函数相容的光滑对称势，多阱势下该唯一性不成立，为构形统计提供了新几何视角。
AI 中文摘要
构形配分函数是经典平衡统计力学中的核心对象，但其逆特征刻画在很大程度上仍未被探索。在本文中，我们研究识别能产生简谐振子高斯构形配分函数的光滑一维势的逆问题。在适当的容许条件下，我们证明配分函数唯一确定由该势生成的推前测度，并通过其逆分支间的简单关系推导得到等价特征刻画。我们建立了一个刚性结果，表明简谐振子是唯一与高斯构形配分函数相容的光滑对称单阱势。我们还证明，对于多阱势，该唯一性会因逆分支的可能重排产生相同测度而丧失。我们的结果揭示了构形统计背后的几何结构，并为统计力学中的逆问题提供了新视角。
英文摘要
Given a configurational partition function, in this work we investigate the inverse problem of reconstructing the one-dimensional potential. For the case of the Gaussian partition function, the configurational density of states (CDOS) is univocally determined, being the harmonic potential uniquely recovered due to its symmetric single--well feature. When multiples inverse branches are considered, the uniqueness of the potential is broken and more information is needed in order to obtain a unique potential. The branch topology imposes alternating orientations, thus allowing distinct spatial realizations. Finally, the Gaussian partition function only determines the CDOS but not the potential, manifesting in this way the precise scope and limitations of the inverse characterization of the Gaussian configurational partition function.
