Mean back relaxation for position and densities

Correlation functions are a standard tool for analyzing statistical particle trajectories. Recently, a so-called mean back relaxation (MBR) has been introduced, which correlates positions at three time points. The deviation of its long-time value from 1/2 has been shown to be a marker for breakage of time-reversal symmetry for confined particles. Here, we extend the analysis of MBR in several ways, including discussion of a cutoff length used when evaluating MBR from trajectory data. Using a path integral approach, we provide a general expression for MBR in terms of multipoint density correlations. For Gaussian systems, this expression yields a relation between MBR and mean-squared displacement. We finally demonstrate that MBR can be applied to other stochastic observables besides particle position. Using it for microscopic densities, its deviation from 1/2 is a marker for broken detailed balance in confinement or in bulk systems.