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 expectation value v and standard deviation $$\delta v$$ of empirical variance $$v[\mathbf {x}]$$ finite series $$\mathbf {x}$$ . is averaged over a fluctuating field $$\sigma _{\mathbf{r}}$$ ( $$\mathbf{r}$$ microcell position) characterized by quenched spatially correlated Gaussian $$g_{\mathbf{r}}$$ Due to -field v(\varDelta \tau )$$ becomes constant, $$\varDelta _{\mathrm {ne}}> 0$$ , large sampling times $$ The volume dependence non-ergodicity parameter {ne}}$$ investigated different spatial correlations. Models with marginally long-ranged -correlations are successfully mapped shear stress data from simulated amorphous glasses polydisperse beads.