Fluctuations of non-ergodic stochastic processes

We investigate the standard deviation [Formula: see text] of the variance [Formula: see text] of time series [Formula: see text] measured over a finite sampling time [Formula: see text] 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 [Formula: see text] must be distinguished: the total variance [Formula: see text] and its contributions [Formula: see text], the typical internal variance within the meta-basins, and [Formula: see text], 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 [Formula: see text] and [Formula: see text] are manifested by their different system-size dependences.