软物质平衡力结构的 Noether 不变性理论
Noether invariance theory for the equilibrium force structure of soft matter
AI总结：
本文详细推导了表征非均匀经典多体系统空间平衡结构的 Noether 不变性理论，揭示了位移场二阶项下两体密度分布的 Hessian 与热平均力关联及力梯度的平衡关系，并通过多种流体和相态的计算机模拟验证了求和规则的有效性。
摘要：
我们详细阐述并推导了 Noether 不变性理论，该理论表征非均匀经典多体系统的空间平衡结构，正如近期针对体系统所提出并研究的 [F. Sammüller 等人, Phys. Rev. Lett. 130, 268203 (2023)]。由此，基于不变变分的 Noether 定理，利用了针对相空间上局部平移变换的内禀热对称性。我们考虑了在参数化该变换的位移场二阶项中出现的平移后果。以自然的方式生成了标准两体密度分布。其空间二阶导数（Hessian）由此与另外两个不同的两体关联函数相平衡，这两个函数分别以系统且微观锐利的方式将热平均力关联和力梯度引入该框架。独立的精确自和异求和规则成立，表达了这种平衡。我们基于计算机模拟验证了该理论的有效性，模拟对象包括 Lennard-Jones 气体、液体和晶体，Weeks-Chandler-Andersen 流体，环境条件下的单原子 Molinero-Moore 水，三体相互作用胶体凝胶形成体，Yukawa 和软球偶极流体，以及 Gay-Berne 粒子的各向同性和向列相。我们明确描述了基于 Noether 定理的求和规则推导，并给出了遵循 Yvon 定理的基于相空间部分积分的更基础证明。
英文摘要：
We give details and derivations for the Noether invariance theory that characterizes the spatial equilibrium structure of inhomogeneous classical many-body systems, as recently proposed and investigated for bulk systems [F. Sammüller $\textit{et al.}$, Phys. Rev. Lett. $\textbf{130}$, 268203 (2023)]. Thereby an intrinsic thermal symmetry against a local shifting transformation on phase space is exploited on the basis of the Noether theorem for invariant variations. We consider the consequences of the shifting that emerge at second order in the displacement field that parameterizes the transformation. In a natural way the standard two-body density distribution is generated. Its second spatial derivative (Hessian) is thereby balanced by two further and different two-body correlation functions, which respectively introduce thermally averaged force correlations and force gradients in a systematic and microscopically sharp way into the framework. Separate exact self and distinct sum rules hold expressing this balance. We exemplify the validity of the theory on the basis of computer simulations for the Lennard-Jones gas, liquid, and crystal, the Weeks-Chandler-Andersen fluid, monatomic Molinero-Moore water at ambient conditions, a three-body-interacting colloidal gel former, the Yukawa and soft-sphere dipolar fluids, and for isotropic and nematic phases of Gay-Berne particles. We describe explicitly the derivation of the sum rules based on Noether's theorem and also give more elementary proofs based on partial phase space integration following Yvon's theorem.
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