Max Planck Institute for the Physics of Complex Systems Noethnitzer Strasse 38 D 01187 Dresden, Germany.
Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel.
Phys Rev E. 2017 Dec;96(6-1):062122. doi: 10.1103/PhysRevE.96.062122. Epub 2017 Dec 15.
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.
近年来，理论和实验都表明，在某些表现出异常扩散的系统中，时间和整体平均均方位移显著不同。整体平均扩散系数是通过标度格林-库伯关系获得的，该关系将标度不变的非平稳速度相关函数与输运系数联系起来。在这里，我们获得了通常在单粒子跟踪实验中记录的时间平均扩散率与潜在的标度不变速度相关函数之间的关系。时间平均均方位移由〈δ^2[over ¯]〉∼2Dνt^βΔ^ν-β给出，其中 t 是总测量时间，Δ是滞后时间。这里 ν 是从整体平均测量得到的异常扩散指数〈x^2〉∼t^ν，而β≥-1 标记了动能〈v^2〉∼t^β的增长或下降。因此，我们建立了一个可以从速度相关函数的渐近性质中读取的指数之间的联系，对于输运常数 Dν 也是如此。我们用非平稳标度不变随机和确定性模型展示了我们的结果，从而强调了在整体平均中具有等效行为的系统在时间平均上可能有很大的差异。如果平均动能有限，β=0，〈δ^2[over ¯]〉和〈x^2〉的时间标度是相同的；然而，时间平均输运系数 Dν 与相应的整体平均扩散常数不同。
