发表机构
Research Institute for Interdisciplinary Science, Okayama University（冈山大学跨学科学术研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文阐明有效哈密顿量的参考测度自由度，证明不同EH是同一内在平衡测度的规范表示，并在宏观极限下由规范无关的大偏差率函数主导。
AI 中文摘要
有效哈密顿量（EH），也称为平均力势或自由能景观，广泛用于描述化学和软物质系统中的平衡性质和慢动力学。然而，传统EH被认为定义不当，因为它在坐标变换下不按标量变换。在这里，我们表明这个明显的问题源于参考测度（规范）的隐式变化，相对于该测度定义了平衡概率密度。我们在慢变量流形上内在地表述平衡测度和福克-普朗克方程（FPE），并表明EH是相对于所选规范定义的标量函数。不同的EH，包括传统EH和最近在黎曼形式中提出的依赖于扩散的EH，因此提供了相同内在平衡测度和随机动力学的不同规范表示。我们进一步表明，规范选择具有操作意义：慢变量的不同约束协议选择不同的参考测度。最后，当平衡测度在宏观极限中服从大偏差原理时，与次主导规范相关的EH之间的差异相对于大偏差速度变得次主导。EH的主导部分然后由规范无关的大偏差率函数确定，相应的确定性动力学同样是规范无关的。
英文摘要
The effective Hamiltonian (EH), also referred to as the potential of mean force or free-energy landscape, is widely used to describe equilibrium properties and slow dynamics in chemical and softmatter systems. However, the conventional EH has been argued to be ill-defined because it does not transform as a scalar under a change of coordinates. Here, we show that this apparent problem originates from an implicit change of the reference measure (gauge) with respect to which the equilibrium probability density is defined. We formulate the equilibrium measure and the Fokker--Planck equation (FPE) intrinsically on the manifold of slow variables and show that an EH is a scalar function defined with respect to a chosen gauge. Different EHs, including the conventional EH and the diffusion-dependent EH recently proposed in a Riemannian formulation, thus provide different gauge representations of the same intrinsic equilibrium measure and stochastic dynamics. We further show that the gauge choice has an operational meaning: different restraint protocols for the slow variables select different reference measures. Finally, when the equilibrium measure obeys a large-deviation principle in a macroscopic limit, differences among EHs associated with subleading gauges become subleading relative to the large-deviation speed. The leading part of the EH is then determined by the gauge-independent large-deviation rate function, and the corresponding deterministic dynamics is likewise gauge independent.
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