Time-Average Based on Scaling Law in Anomalous Diffusions

To solve the obscureness in measurement brought about from the weak ergodicity breaking appeared in anomalous diffusions we have suggested the time-averaged mean squared displacement (MSD) $\bar{\delta^2 (\tau)}_\tau$ with a integral interval depending linearly on the lag time $\tau$. For the continuous time random walk describing a subdiffusive behavior, we have found that $\bar{\delta^2 (\tau)}_\tau \sim \tau^\gamma$ like that of the ensemble-averaged MSD, which makes it be possible to measure the proper exponent values through time-average in experiments like a single molecule tracking. Also we have found that it is originated from the scaling nature of the MSD at a aging time in anomalous diffusion and confirmed them through numerical results of the other microscopic non-Markovian model showing subdiffusions and superdiffusions with the origin of memory enhancement.