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

TL;DR

This paper introduces a new method to estimate the steady-state diffusion coefficient from transient regime data, enabling analysis without requiring the system to reach equilibrium, validated through simulations of lattice systems.

Contribution

A novel approach to determine the steady-state diffusion coefficient using early transient data, bypassing the need for steady-state achievement.

Findings

Accurately estimates the critical transition time $t^*$ from short-time data.

Provides reliable steady-state diffusion coefficients without reaching equilibrium.

Validated method on various two-dimensional lattice systems.

Abstract

When particles/molecules diffuse in systems that contain obstacles, the steady-state regime (during which the mean-square displacement scales linearly with time, $\left< r^2 \right> \sim t$) 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 $\left< r^2 \right> \sim t^\alpha$, where the corresponding exponent $\alpha<1$. We propose a new method to estimate the critical time $t^*$ that marks the transition between these two regimes. The method uses short-time data from the transient regime to estimate $t^*$, 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.