We investigate the standard deviation δv(∆t) of the variance v[x] of time series x measured over a finite sampling time ∆t 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[x ck ] must be distinguished: the total variance δv 2 tot = δv 2 int +δv 2 ext and its contributions δv 2 int , the typical internal variance within the meta-basins, and δv 2 ext , 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 δvint and δvext are manifested by their different system-size dependences.

Using molecular dynamics simulation of a polymer glass model we investigate free-standing polymer films focusing on the in-plane shear modulus µ, defined by means of the stress-fluctuation formula, as a function of temperature T , film thickness H (tuned by means of the lateral box size L) and sampling time ∆t. Various observables are seen to vary linearly with 1/H demonstrating thus the (to leading order) linear superposition of bulk and surface properties. Confirming the timetranslational invariance of our systems, µ(∆t) is shown to be numerically equivalent to a second integral over the shear-stress relaxation modulus G(t). It is thus a natural smoothing function statistically better behaved as G(t). As shown from the standard deviations δµ and δG, this is especially important for large times and for temperatures around the glass transition. µ and G are found to decrease continuously with T and a jump-singularity is not observed. Using the Einstein-Helfand relation for µ(∆t) and the successful time-temperature superposition scaling of µ(∆t) and G(t) the shear viscosity η(T ) can be estimated for a broad range of temperatures.

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