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 〈δ^{2}[over ¯]〉∼2D_{ν}t^{β}Δ^{ν-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 〈x^{2}〉∼t^{ν}, while β≥-1 marks the growth or decline of the kinetic energy 〈v^{2}〉∼t^{β}. 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_{ν}. 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 〈δ^{2}[over ¯]〉 and 〈x^{2}〉 are identical; however, the time-averaged transport coefficient D_{ν} is not identical to the corresponding ensemble-averaged diffusion constant.