Ensemble fluctuations matter for variances of macroscopic variables

Extending recent work on stress fluctuations in complex fluids and amorphous solids we describe in general terms the ensemble average v ( Δ t ) and the standard deviation δ v ( Δ t ) of the variance v [ x ] of time series x of a stochastic process x ( t ) measured over a finite sampling time Δ t . Assuming a stationary, Gaussian and ergodic process, δ v is given by a functional δ v G [ h ] of the autocorrelation function h ( t ). δ v ( Δ t ) is shown to become large and similar to v ( Δ t ) if Δ t corresponds to a fast relaxation process. Albeit δ v = δ 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. Graphic abstract