Fluctuations of non-ergodic stochastic processes

TL;DR

This paper analyzes the fluctuations of variance in non-ergodic stochastic processes, emphasizing the importance of configuration and time averaging order, and distinguishes between internal and external variance contributions in physical systems.

Contribution

It introduces a detailed framework for understanding variance fluctuations in non-ergodic systems, including the distinction between internal and external variances and their size dependencies.

Findings

Variance fluctuations depend on the order of averaging over configurations and time.

Internal and external variances show different system-size dependencies.

The framework is applied to shear-stress fluctuations in various disordered materials.

Abstract

We investigate the standard deviation $\delta v(\tsamp)$ of the variance $v[\xbf]$ of time series $\xbf$ measured over a finite sampling time $\tsamp$ focusing on non-ergodic systems where independent "configurations" $c$ get trapped in meta-basins of a generalized phase space. It is thus relevant in which order averages over the configurations $c$ and over time series $k$ of a configuration $c$ are performed. Three variances of $v[\xbf_{ck}]$ must be distinguished: the total variance $\dvtot = \dvint + \dvext$ and its contributions $\dvint$, the typical internal variance within the meta-basins, and $\dvext$, characterizing the dispersion between the different basins. We discuss simplifications for physical systems where the stochastic variable $x(t)$ is due to a density field averaged over a large system volume $V$. The relations are illustrated for the shear-stress fluctuations in quenched elastic networks and low-temperature glasses formed by polydisperse particles and free-standing polymer films. The different statistics of $\svint$ and $\svext$ are manifested by their different system-size dependence