Fluctuations of non-ergodic stochastic processes

We investigate the standard deviation δ v ( Δ t ) of the variance v [ x ] of time series x measured over a finite sampling time Δ t 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 [ x ck ] must be distinguished: the total variance δ v tot 2 = δ v int 2 + δ v ext 2 and its contributions δ v int 2 , the typical internal variance within the meta-basins, and δ v ext 2 , 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 δ v int and δ v ext are manifested by their different system-size dependences. Graphic abstract