Spatial correlation functions for non-ergodic stochastic processes of macroscopic system

TL;DR

This paper analyzes the variances of time averages in non-ergodic macroscopic systems, expressing them as volume averages of correlation functions, and illustrates the approach with lattice spring models.

Contribution

It introduces a method to decompose variances into internal and external correlation functions for non-ergodic systems, linking variances to spatial correlations.

Findings

Total variance decomposes into internal and external parts.

Correlation functions can be used to trace variance dependencies.

Application to lattice spring models demonstrates the approach.

Abstract

Focusing on non-ergodic macroscopic systems we reconsider the variances of time averages time-series. The total variance (direct average over all time-series) is known to be the sum of an internal variance (fluctuations within the meta-basins) and an external variance (fluctuations between meta-basins). It is shown that whenever the time-averaged observable can be expressed as a volume average of a local field the three variances can be written as volume averages of correlation functions with the total correlation function being the sum of an internal and an external correlation function. The dependences of the the different variancescan thus be traced back to the internal and the external correlation function. Various relations are illustrated using lattice spring models with spatially correlated spring constants.