发表机构
Universität Bayreuth（拜罗伊特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出多元超密度泛函理论，将其与机器学习结合，通过受限硬棒流体聚类案例验证，可预测超可观测量诱导的统计量。
AI 中文摘要
超密度泛函理论可用于研究空间非均匀经典多体系统中一般序参量或统计力学可观测量的平衡行为，该方法基于将精确的Mermin-Evans经典密度泛函映射应用于扩展系综。本文提出了一种多元泛化方法，用于同时研究所选的多种不同超可观测量的性质及其相互关联，所得框架为一般多体现象提供了系统的表征和预测方案。我们明确证明，所有相关的平衡平均、方差和协方差均构成普适密度泛函；相关的单体超涨落剖面可量化局域密度与超可观测量的一阶、二阶组合的关联程度，这些多元超涨落剖面可在多体模拟中获取，且满足我们从扩展多元系综的最小化原理导出的精确超Ornstein-Zernike方程。该理论的形式结构可自然地与监督机器学习结合，使所有超密度泛函可通过在模拟数据上训练神经网络在实践中获取。我们以受限硬棒流体的聚类为例演示所有关键技术，选取总粒子数和最大团簇尺寸作为代表性超可观测量，数值方法可高效且成功地预测所选超可观测量诱导的所有统计量，我们通过与测试数据的比较验证了这一点，并将其归因于该通用方法结合的第一性原理与机器学习概念的紧密协同。
英文摘要
Hyperdensity functional theory facilitates the investigation of the equilibrium behavior of a general order parameter or statistical mechanical observable in spatially inhomogeneous classical many-body systems. The approach is based on applying the exact Mermin-Evans classical density functional mapping to an extended ensemble. Here we present the multivariate generalization for investigating simultaneously the properties and interrelations of several different hyperobservables of choice. The resulting framework gives rise to a systematic characterization and prediction scheme for general many-body phenomena. All pertinent equilibrium averages, variances, and covariances constitute universal density functionals, as we demonstrate explicitly. Associated one-body hyperfluctuation profiles quantify the degree of correlation of the local density with first- and second-order combinations of hyperobservables. These multivariate hyperfluctuation profiles are accessible in many-body simulations and they satisfy exact hyper-Ornstein-Zernike equations, which we derive from the minimization principle in the extended multivariate ensemble. The formal structure of the theory integrates naturally with supervised machine learning, which renders all hyperdensity functionals accessible in practice via training of neural networks on simulation data. We demonstrate all salient techniques using the illustrative case of clustering in confined hard rod fluids, thereby choosing the total number of particles and the largest cluster size as the representative hyperobservables of interest. Our numerical methodology enables the efficient and successful prediction of all statistical quantities induced by the chosen hyperobservables, which we verify via comparison to test data and which we attribute to the tight interplay of first-principles and machine-learning concepts that our general approach combines.
Comments20 pages, 4 figures
