宏观响应诊断终态结果的噪声敏感性

中文摘要 AI 辅助

能否在不模拟完整轨迹的情况下，从微观初始数据推断多体系统的宏观终态结果？我们并未构建这种捷径，而是针对高斯微观输入探讨一个更基本的问题：随着系统规模增长，任何固定的Wiener-Hermite阶数能否保留终态结果方差的非零比例？我们考虑具有独立高斯无序的均匀系统，其中所有微观坐标在对称性上等价，终态事件对每个无序变量单调，且均匀无序平移与控制场平移精确等价，并具有与系统规模无关的转换因子。利用正向和逆向高斯影响界以及高斯Russo公式，我们推导出一个可直接测量的判据，该判据是噪声敏感性的充分必要条件。具体而言，当且仅当在平衡阈值处结果概率相对于控制场的斜率增长慢于系统体积的平方根时，原始终态结果与坐标扰动后的终态结果之间的相关性对于每个固定的非零坐标噪声水平渐近消失。对线性尺寸达192的三维驱动随机场Ising模型的事件驱动模拟发现，归一化响应和扰动样本间的相关性整体下降，与有限尺寸下的噪声敏感区域一致。对于具有预设种子的空间Stag-Hunt动力学，该判据通过有效影响坐标数进行推广。此外，对路径依赖的最佳响应博弈的模拟显示，在所研究的尺寸范围内，终态均衡可能依赖于更新顺序，同时仍携带大量的有限阶预测信息。

英文摘要

Can the terminal macroscopic outcome of a many-body system be inferred from its microscopic initial data without simulating the full trajectory? Rather than construct such a shortcut, we address a more fundamental question for Gaussian microscopic inputs: can any fixed Wiener-Hermite degree retain a nonvanishing fraction of the variance of the terminal outcome as the system grows? We consider homogeneous systems with independent Gaussian disorder in which all microscopic coordinates are symmetry-equivalent, the terminal event is monotone in each disorder variable, and a uniform disorder shift is exactly equivalent to a control-field shift with a size-independent conversion factor. Using forward and inverse Gaussian influence bounds together with a Gaussian Russo formula, we derive a directly measurable criterion that is both necessary and sufficient for noise sensitivity. Specifically, the correlation between the original and coordinate-perturbed terminal outcomes vanishes asymptotically for every fixed nonzero level of coordinatewise noise if and only if the slope of the outcome probability with respect to the control field at the balanced threshold grows more slowly than the square root of the system volume. Event-driven simulations of the three-dimensional driven random-field Ising model up to linear size 192 find that both the normalized response and the correlations between perturbed samples decrease overall, consistent with the noise-sensitive regime at finite size. For spatial Stag-Hunt dynamics with prescribed seeds, the criterion generalizes through an effective number of influential coordinates. Separately, simulations of a path-dependent best-response game show, over the sizes studied, that the terminal equilibrium can depend on the update schedule while still carrying substantial finite-order predictive information.

发表机构

• Wuhan City Polytechnic（武汉城市职业学院）
• School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

19 pages, 6 figures