Fluctuations of non-ergodic stochastic processes

G George and L Klochko and AN Semenov and J Baschnagel and JP Wittmer, EUROPEAN PHYSICAL JOURNAL E, 44, 54 (2021).

DOI: 10.1140/epje/s10189-021-00070-5

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

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