Estimating the Steady State Diffusion Coefficient Using Data from the Transient Anomalous Regime

Abstract

When particles/molecules diffuse in systems that contain obstacles, the steady-state regime (during which the mean-square displacement scales linearly with time, $⟨r^2⟩\sim t$) is preceded by a transient regime. It is common to characterize this transient regime using the concept of anomalous (sub)diffusion with the scaling law $⟨r^2⟩\sim t^{\alpha }$, where the corresponding exponent $\alpha <1$. We propose a new method to estimate the critical time $t^{\ast }$ that marks the transition between these two regimes. The method uses short-time data from the transient regime to estimate $t^{\ast }$, which can then be used to estimate the steady-state diffusion coefficient $D$. In other words, we propose a procedure that makes it possible to estimate the steady state diffusion coefficient without reaching the steady-state. We test the procedure with various two-dimensional lattice systems.