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 <(delta) over bar (2)> 2D(v)t beta Delta t beta, where t is the total measurement time and Delta is the lag time. Here. is the anomalous diffusion exponent obtained from ensemble-averaged measurements < x(2)> similar to t(v), while beta >= -1 marks the growth or decline of the kinetic energy < v2 > similar to t(beta). 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 D.. 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, beta = 0, the time scaling of <(delta) over bar (2)> and < x(2)> are identical; however, the time-averaged transport coefficient D. is not identical to the corresponding ensemble-averaged diffusion constant.