Fluctuations of non-ergodic stochastic processes

We investigate the standard deviation $delta v(tsamp)$ of the variance $v[xbf]$ of time series $xbf$ measured over a finite sampling time $tsamp$ 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 $v[xbf_{ck}]$ must be distinguished: the total variance $dvtot = dvint + dvext$ and its contributions $dvint$, the typical internal variance within the meta-basins, and $dvext$, 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 $svint$ and $svext$ are manifested by their different system-size dependence