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 $\langle \overline{{\delta }^2}\rangle \sim 2D_{\nu }{t}^{\beta }{\mathrm{\Delta }}^{\nu -\beta }$, where $t$ is the total measurement time and $\mathrm{\Delta }$ is the lag time. Here $\nu$ is the anomalous diffusion exponent obtained from ensemble-averaged measurements $\langle x^2\rangle \sim t^{\nu }$, while $\beta \ge -1$ marks the growth or decline of the kinetic energy $\langle v^2\rangle \sim t^{\beta }$. 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 $D_{\nu }$. 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 $\langle \overline{{\delta }^2}\rangle$ and $\langle x^2\rangle$ are identical; however, the time-averaged transport coefficient $D_{\nu }$ is not identical to the corresponding ensemble-averaged diffusion constant.

• Received 28 August 2017

DOI:https://doi.org/10.1103/PhysRevE.96.062122