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

We investigate simple models for strictly non-ergodic stochastic processes x t ( t being the discrete time step) focusing on the expectation value v and the standard deviation δ v of the empirical variance v [ x ] of finite time series x . x t is averaged over a fluctuating field σ r ( r being the microcell position) characterized by a quenched spatially correlated Gaussian field g r . Due to the quenched g r -field δ v ( Δ τ ) becomes a finite constant, Δ ne > 0 , for large sampling times Δ τ . The volume dependence of the non-ergodicity parameter Δ ne is investigated for different spatial correlations. Models with marginally long-ranged g r -correlations are successfully mapped on shear stress data from simulated amorphous glasses of polydisperse beads.