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 average mean squared displacement are remarkably different. The ensemble average diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale invariant non-stationary 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 $\overline{δ^2} \sim 2 D_νt^βΔ^{ν-β}$ where $t$ is the total measurement time and $Δ$ the lag time. Here $ν>1$ is the anomalous diffusion exponent obtained from ensemble averaged measurements $\langle x^2 \rangle \sim t^ν$ while $β\ge -1$ marks the growth or decline of the kinetic energy $\langle v^2 \rangle \sim t^β$. Thus we establish a connection between exponents which 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 non-stationary scale invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. This is the case, for example, if averaged kinetic energy is finite, i.e. $β=0$, where $\langle \overline{δ^2}\rangle \neq \langle x^2\rangle$.