Simple models for strictly non-ergodic stochastic processes of macroscopic systems.

We investigate simple models for strictly non-ergodic stochastic processes [Formula: see text] (t being the discrete time step) focusing on the expectation value v and the standard deviation [Formula: see text] of the empirical variance [Formula: see text] of finite time series [Formula: see text]. [Formula: see text] is averaged over a fluctuating field [Formula: see text] ([Formula: see text] being the microcell position) characterized by a quenched spatially correlated Gaussian field [Formula: see text]. Due to the quenched [Formula: see text]-field [Formula: see text] becomes a finite constant, [Formula: see text], for large sampling times [Formula: see text]. The volume dependence of the non-ergodicity parameter [Formula: see text] is investigated for different spatial correlations. Models with marginally long-ranged [Formula: see text]-correlations are successfully mapped on shear stress data from simulated amorphous glasses of polydisperse beads.