Evaluating non-equilibrium trajectories via mean back relaxation: Dependence on length and time scales

The mean back relaxation (MBR) relates the value of a stochastic process at three different time points. It has been shown to detect broken detailed balance under certain conditions. For experiments of probe particles in living and passivated cells, MBR was found to be related to the so-called effective energy, which quantifies the violation of the fluctuation-dissipation theorem. In this paper, we discuss the dependence on the length and time parameters that enter MBR, both for cells as well as for a model system, finding qualitative agreement between the two. For the cell data, we extend the phenomenological relation between MBR and effective energy to a larger range of time parameters compared to previous work, allowing us to test it in systems with limited resolution. We analyze the variance of back relaxation (VBR) in dependence of the mentioned parameters, relevant for the statistical error in MBR evaluation. For Gaussian systems, the variance is found analytically in terms of the mean squared displacement, and we determine its absolute minimum as a function of the length and time parameters. Comparing VBR from cell data to a Gaussian prediction demonstrates a non-Gaussian process.