Random walk on a lattice in the presence of obstacles: The short-time transient regime, anomalous diffusion and crowding

Abstract

The diffusion of a particle in a crowded environment typically proceeds through three regimes: for very short times the particle diffuses freely until it collides with an obstacle for the first time, while for very long times diffusion the motion is Fickian with a diffusion coefficient $D$ that depends on the concentration and type of obstacles present in the system. For intermediate times, the mean-square displacement of the particle often increases approximately as $~t^\alpha$, with $\alpha<1$, typical of what is generally called anomalous diffusion. However, it is not clear how one can identify or choose a time or displacement interval that would give a reliable estimate of $\alpha$. In this paper, we use two exact numerical approaches to obtain diffusion data for a simple Lattice Monte Carlo model in both time limits. This allows us to propose an objective definition of the transient regime and a unique value for $\alpha$. Furthermore, our methodology directly gives us the length scale over which the transient regime switches to the steady-state regime. We test our proposed approach using several types of obstacle systems, and we introduce the novel concept of excess diffusion lengths. Finally, we show that the values of the parameters describing the anomalous transient regime depend on the Monte Carlo moves used to describe the dynamics of the particle, and we propose a new algorithm that correctly models the short time diffusion of a particle on a lattice.