Ensemble fluctuations matter for variances of macroscopic variables

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.