Correlated measurements in single-particle-tracking dynamics

We investigate the time-averaged displacement correlation function in single-particle tracking within the generalized Langevin equation framework, reformulated in terms of the velocity correlation function. We identify scaling laws for the normalized displacement correlation function applicable to both nonstationary and aging dynamics. This correlation decays more rapidly with lag time in aging dynamics than in nonaging but nonstationary process. We further examine the underlying ergodic properties of the process and introduce a kinetic energy-based metric to define the effective friction. As an application, we analyze a multiexponential memory kernel commonly used in the protein-folding studies and show that the effective friction increases when components with a shorter decay time are incorporated.