Time-Average Based on Scaling Law in Anomalous Diffusions

TL;DR

This paper introduces a new time-averaged mean squared displacement method for analyzing anomalous diffusions, enabling accurate measurement of diffusion exponents despite weak ergodicity breaking, supported by theoretical and numerical evidence.

Contribution

It proposes a time-average based on a linear integral interval that accurately reflects ensemble-averaged MSD in anomalous diffusion, addressing measurement obscurities.

Findings

Time-averaged MSD scales with lag time similarly to ensemble MSD.

The method works for subdiffusive and superdiffusive behaviors.

Numerical results confirm the scaling origin from the MSD at aging time.

Abstract

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.