Abstract

Extending recent work on stress fluctuations in complex fluids and amorphous solids we describe in general terms the ensemble average \(v(\Delta t)\) and the standard deviation \(\delta v(\Delta t)\) of the variance \(v[{\mathbf {x}}]\) of time series \({\mathbf {x}}\) of a stochastic process x(t) measured over a finite sampling time \(\Delta t\). Assuming a stationary, Gaussian and ergodic process, \(\delta v\) is given by a functional \(\delta v_{\mathrm {G}}[h]\) of the autocorrelation function h(t). \(\delta v(\Delta t)\) is shown to become large and similar to \(v(\Delta t)\) if \(\Delta t\) corresponds to a fast relaxation process. Albeit \(\delta v = \delta v_{\mathrm {G}}[h]\) does not hold in general for non-ergodic systems, the deviations for common systems with many microstates are merely finite-size corrections. Various issues are illustrated for shear-stress fluctuations in simple coarse-grained model systems.

Graphic abstract

中文翻译：

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整体波动对宏观变量的方差很重要

摘要

扩展了关于复杂流体和无定形固体中应力波动的最新工作，我们以一般术语描述了方差\的集合平均\（v（\ Delta t）\）和标准偏差\（\ delta v（\ Delta t）\）。在有限采样时间\（\ Delta t \）上测量的随机过程x（t）的时间序列\（{\ mathbf {x}} \）的（v [{\ mathbf {x}}] \）。假设平稳，高斯和遍历过程，\（\ delta v \）由自相关函数h（t）的函数\（\ delta v _ {\ mathrm {G}} [h] \）给出。\（\ delta v（\ Delta t）\）如果\（\ Delta t \）对应于快速弛豫过程，则表示变大并类似于\（v（\ Delta t）\）。尽管\（\ delta v = \ delta v _ {\ mathrm {G}} [h] \）通常对于非遍历系统不成立，但具有许多微状态的常见系统的偏差仅是有限大小的校正。在简单的粗粒度模型系统中，为剪切应力的波动说明了各种问题。

图形摘要