Eur. Phys. J. E (2022) 45: 65
https://doi.org/10.1140/epje/s10189-022-00222-1

Regular Article - Flowing Matter

Different types of spatial correlation functions for non-ergodic stochastic processes of macroscopic systems

Institut Charles Sadron, Université de Strasbourg & CNRS, 23 rue du Loess, 67034, Strasbourg Cedex, France

^a joachim.wittmer@ics-cnrs.unistra.fr

Received: 2 June 2022
Accepted: 27 July 2022
Published online: 6 August 2022

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 $\delta {\mathcal{O}}_{\mathrm{tot}}^2=\delta {\mathcal{O}}_{\mathrm{int}}^2+\delta {\mathcal{O}}_{\mathrm{ext}}^2$ (direct average over all time series) is known to be the sum of an internal variance $\delta {\mathcal{O}}_{\mathrm{int}}^2$ (fluctuations within the meta-basins) and an external variance $\delta {\mathcal{O}}_{\mathrm{ext}}^2$ (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 $\Delta \tau$ 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.