Different types of spatial correlation functions for non-ergodic stochastic processes of macroscopic systems

Focusing on non-ergodic macroscopic systems, we reconsider the variances [Formula: see text] of time averages [Formula: see text] of time-series [Formula: see text]. The total variance [Formula: see text] (direct average over all time series) is known to be the sum of an internal variance [Formula: see text] (fluctuations within the meta-basins) and an external variance [Formula: see text] (fluctuations between meta-basins). It is shown that whenever [Formula: see text] can be expressed as a volume average of a local field [Formula: see text] the three variances can be written as volume averages of correlation functions [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text]. The dependences of the [Formula: see text] on the sampling time [Formula: see text] and the system volume V can thus be traced back to [Formula: see text] and [Formula: see text]. Various relations are illustrated using lattice spring models with spatially correlated spring constants. .