Kinetic Energy Diffusivity and Scaling Velocity Correlation Functions

We propose a Green-Kubo-like relation for kinetic energy diffusivity to investigate the interplay between ergodicity and anomalous diffusion. This approach introduces a fluctuation metric for the time-averaged kinetic energy, which holds for scaling velocity correlation functions. We demonstrate that as stationary diffusive systems transition into an effective ergodic phase, their kinetic energy metric converges to a universal law. This finding provides a robust framework for understanding the dynamics of such systems. Applications to protein folding and single-particle tracking illustrate the practical utility of our approach, offering a clear prescription for extracting key physical parameters, such as the friction constant and relaxation time, from finite experimental datasets. Importantly, this method remains effective even when the underlying processes exhibit weak ergodicity breaking or are bounded. Furthermore, we explore the nonergodic transition associated with the aging velocity correlation function observed in granular gases.