Abstract

Focusing on non-ergodic macroscopic systems, we reconsider the variances \(\delta \mathcal{O}^2\) of time averages \(\mathcal{O}[\mathbf {x}]\) of time-series \(\mathbf {x}\). The total variance \(\delta \mathcal{O}^2_{\mathrm {tot}}= \delta \mathcal{O}^2_{\mathrm {int}}+ \delta \mathcal{O}^2_{\mathrm {ext}}\) (direct average over all time series) is known to be the sum of an internal variance \(\delta \mathcal{O}^2_{\mathrm {int}}\) (fluctuations within the meta-basins) and an external variance \(\delta \mathcal{O}^2_{\mathrm {ext}}\) (fluctuations between meta-basins). It is shown that whenever \(\mathcal{O}[\mathbf {x}]\) can be expressed as a volume average of a local field \(\mathcal{O}_{\mathbf{r}}\) the three variances can be written as volume averages of correlation functions \(C_{\mathrm {tot}}(\mathbf{r})\), \(C_{\mathrm {int}}(\mathbf{r})\) and \(C_{\mathrm {ext}}(\mathbf{r})\) with \(C_{\mathrm {tot}}(\mathbf{r}) = C_{\mathrm {int}}(\mathbf{r}) + C_{\mathrm {ext}}(\mathbf{r})\). The dependences of the \(\delta \mathcal{O}^2\) on the sampling time \(\varDelta \tau \) and the system volume V can thus be traced back to \(C_{\mathrm {int}}(\mathbf{r})\) and \(C_{\mathrm {ext}}(\mathbf{r})\). Various relations are illustrated using lattice spring models with spatially correlated spring constants.

Graphical abstract

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中文翻译：

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宏观系统非遍历随机过程的不同类型空间相关函数

摘要

关注非遍历宏观系统，我们重新考虑时间序列\ (\mathcal{O}[\mathbf {x}]\)的时间平均值的方差\(\delta \mathcal{O}^2\) \mathbf {x}\)。总方差\(\delta \mathcal{O}^2_{\mathrm {tot}}= \delta \mathcal{O}^2_{\mathrm {int}}+ \delta \mathcal{O}^2_{\ mathrm {ext}}\)（所有时间序列的直接平均值）已知是内部方差的总和\(\delta \mathcal{O}^2_{\mathrm {int}}\)（元数据中的波动-盆地）和外部方差\(\delta \mathcal{O}^2_{\mathrm {ext}}\)（元盆地之间的波动）。结果表明，每当\(\mathcal{O}[\mathbf {x}]\)可以表示为局部场的体积平均值\(\mathcal{O}_{\mathbf{r}}\)这三个方差可以写成相关函数的体积平均值\(C_{\mathrm {tot}} (\mathbf{r})\) , \(C_{\mathrm {int}}(\mathbf{r})\)和\(C_{\mathrm {ext}}(\mathbf{r})\)与\(C_{\mathrm {tot}}(\mathbf{r}) = C_{\mathrm {int}}(\mathbf{r}) + C_{\mathrm {ext}}(\mathbf{r})\ ) . \(\delta \mathcal{O}^2\)对采样时间\(\varDelta \tau \)和系统体积V的依赖关系因此可以追溯到\(C_{\mathrm {int}} (\mathbf{r})\)和\(C_{\mathrm {ext}}(\mathbf{r})\). 使用具有空间相关弹簧常数的晶格弹簧模型来说明各种关系。

图形概要

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