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

Abstract

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 $\delta v$ of the empirical variance $v[x]$ of finite time series $x$. $x_t$ is averaged over a fluctuating field $\sigma_{r}$ ($r$ being the microcell position) characterized by a quenched spatially correlated Gaussian field. Due to the quenched field $\delta v(\Delta t)$ becomes a finite constant, $\Delta_{ne} > 0$, for large sampling times $\Delta t$. The volume dependence of the non-ergodicity parameter $\Delta_{ne}$ is investigated for different spatial correlations. Models with marginally long-ranged $\fr$-correlations are successfully mapped on shear-stress data from simulated amorphous glasses of polydisperse beads.