Ensemble fluctuations matter for variances of macroscopic variables

TL;DR

This paper analyzes how ensemble fluctuations influence the variance of macroscopic variables in stochastic processes, especially in complex fluids and amorphous solids, highlighting the importance of ensemble effects and finite-size corrections.

Contribution

It generalizes the understanding of ensemble fluctuations in variances of time series, extending previous work to include non-ergodic systems and finite-size effects.

Findings

elta v(\u0394 t) becomes large near fast relaxation times

Finite-size corrections are significant in non-ergodic systems with many microstates

Ensemble fluctuations impact stress variance measurements in complex fluids

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 $\delta v_G[h]$ 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 $\delta v = \delta v_G[h]$ 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.