Abstract

The connection between ergodicity breaking for a forced system and its two-time velocity correlation function is demonstrated using the Khinchin theorem, where the particle dynamics is described by a generalized Langevin equation. Products of observables at a pair of time points are constructed that are correlated with their initial preparations. Three types of nonergodic behaviors are elucidated, specifically the velocity correlation function of the system either approaches a constant, oscillates with time around a non-vanishing constant or oscillates around zero-value at large times. The corresponding noise spectral densities behave as low-hindering and high-passing, band-passing as well as low-passing and high-hindering or simply band-hindering. When a harmonic potential is imposed on the system, the first situation can be transformed into ergodicity whereas the latter two types cannot. Furthermore, the application to the Debye Brownian oscillator as well as the verification of several famous models are discussed.

Graphical abstract

中文翻译：

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通过频谱分析解开遍历性分解的根源

摘要

使用Khinchin定理证明了强迫系统的遍历性破坏与其两次速度相关函数之间的联系，其中，动力学由广义Langevin方程描述。构建在两个时间点的可观察产品，并与其初始准备相关。阐明了三种类型的非遍历行为，特别是系统的速度相关函数要么接近一个常数，要么在不消失的常数附近随时间振荡，要么在较大的时候在零值附近振荡。相应的噪声频谱密度表现为低阻碍和高通，带通以及低通过和高阻碍或简单地带阻。当系统施加谐波电位时，第一种情况可以转换为遍历性，而后两种类型则不能。此外，还讨论了在德拜布朗振荡器中的应用以及几种著名模型的验证。

图形概要