使图拉普拉斯算子物理化：电振荡器网络中的多尺度粗粒化
Making the Graph Laplacian Physical: Multiscale Coarse-Graining in Electrical Oscillator Networks
- Instituto de Tecnología Química (ITQ), Consejo Superior de Investigaciones Científicas-Universitat Politècnica de València（化工技术研究所（ITQ），西班牙高等科学研究委员会-瓦伦西亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明电阻耦合LC网络中图拉普拉斯算子的特征模态对应可测量的电压模式，通过粗粒化将1002个谐振器降为42节点RLC电路，误差仅1.4-4.6%，使谱成为物理设计原则。
AI中文摘要：
谱间隙被广泛解释为集体组织的标志，但所得模态究竟对应于物理变量还是仅仅对应于方便的数学坐标，这一点很少是明确的。在此我们表明，在电阻耦合的LC网络中，图拉普拉斯算子定义了一个实验上可访问的电学描述层级。其特征向量成为可测量的电压模式，而其特征值则量化了耦合引起的这些模式的电阻阻尼。当区域内耦合强而区域间耦合弱时，微观电压坍缩为嵌套的集体变量。一个由1002个谐振器组成的网络首先缩减为42个区域电压，然后缩减为四个由几何控制的全局模态。对微观基尔霍夫方程求和可产生一个可直接构造的42节点RLC电路，无需参数拟合，该电路以1.4%至4.6%的全局均方根误差再现区域动力学。更精细的模态坐标可提高精度，但不再普遍对应于简单的电阻图。鲁棒性测试表明，区域层级在相当大的电导无序和内部重连下仍然存在，但当扰动损害区域内混合时其会减弱；区域间连接性的变化则选择性地破坏全局层级。因此，拉普拉斯谱成为一种物理设计原则：它识别哪些集体电学变量会出现、如何构造它们，以及它们何时能提供充分的降阶描述。
英文摘要：
Spectral gaps are widely interpreted as signatures of collective organization, yet it is rarely clear whether the resulting modes correspond to physical variables or merely to convenient mathematical coordinates. Here we show that, in resistor-coupled LC networks, the graph Laplacian defines an experimentally accessible hierarchy of electrical descriptions. Its eigenvectors become measurable voltage patterns, while its eigenvalues quantify the coupling-induced resistive damping of those patterns. When coupling is strong within regions and weak between them, microscopic voltages collapse into nested collective variables. A network of 1002 resonators reduces first to 42 regional voltages and then to four geometry-controlled global modes. Summing the microscopic Kirchhoff equations produces a directly constructible 42-node RLC circuit, without parameter fitting, that reproduces regional dynamics with 1.4 to 4.6% global RMS error. Finer modal coordinates improve accuracy but no longer correspond generally to simple resistor graphs. Robustness tests show that the regional level survives substantial conductance disorder and internal rewiring, but weakens when perturbations impair mixing within a region; changes in interregional connectivity selectively disrupt the global level. Thus the Laplacian spectrum becomes a physical design principle: it identifies which collective electrical variables emerge, how they can be constructed, and when they provide an adequate reduced description.
