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Correlation functions are a standard tool for analyzing statistical particletrajectories. Recently, a so called mean back relaxation (MBR) has beenintroduced, which correlates positions at three time points and which has beenshown to be a marker for breakage of time reversal symmetry for confinedparticles. Here, we extend this discussion in several ways. We give anextension to multiple dimensions. Using a path integral approach, we provide ageneral expression for MBR in terms of multipoint density correlations. ForGaussian systems, this expression yields a relation between MBR and meansquared displacement. From it, several properties of MBR for Gaussianequilibrium systems can be found, such as monotonicity and positivity.Unexpectedly, in this case the particle forgets the conditioning via a powerlaw in time. We finally demonstrate that using MBR for microscopic densities,it is a marker for broken detailed balance even in bulk systems.

Mean Back Relaxation for Position and Densities