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 $\frac{1}{2}$ 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 cut off 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 $\frac{1}{2}$ is a marker for broken detailed balance in confinement or in bulk systems.