Scale-invariant Green-Kubo relation for time-averaged diffusivity

In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time- and ensemble-averaged mean-squared displacement are remarkably different. The ensemble-averaged diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale-invariant nonstationary velocity correlation function with the transport coefficient. Here we obtain the relation between time-averaged diffusivity, usually recorded in single-particle tracking experiments, and the underlying scale-invariant velocity correlation function. The time-averaged mean-squared displacement is given by (δ2̄)∼2DνtβΔν-β, where t is the total measurement time and Δ is the lag time. Here ν is the anomalous diffusion exponent obtained from ensemble-averaged measurements (x2)∼tν, while β≥-1 marks the growth or decline of the kinetic energy (v2)∼tβ. Thus, we establish a connection between exponents that can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant Dν. We demonstrate our results with nonstationary scale-invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. If the averaged kinetic energy is finite, β=0, the time scaling of (δ2̄) and (x2) are identical; however, the time-averaged transport coefficient Dν is not identical to the corresponding ensemble-averaged diffusion constant.