Discriminating between anomalous diffusion and transient behavior in microheterogeneous environments

Abstract

Diffusion in macrohomogeneous and microheterogeneous media can be described as effective free diffusion only at sufficiently long times. At intermediate times, the mean-square displacement of a diffusing object shows a transient behavior that can be misinterpreted as anomalous subdiffusion. We discuss how to discriminate between the two.

Figure 1

(A) The mean-square displacement 〈Δx²(t)〉 of a particle in the presence of periodically spaced permeable membranes in the direction normal to the membranes. (Inset) System geometry. The parameters in dimensionless units: membrane permeability is 0.01; intermembrane distance is 10; and particle diffusion coefficient in free space is 1. (Dotted and dashed straight lines) Short- and long-time asymptotic behaviors of 〈Δx²(t)〉, respectively. (B) The three exponents α_fit(t), α_int(t), and α_dif(t) obtained by interpreting the transient behavior of 〈Δx²(t)〉 as anomalous subdiffusion. (Inset) The exponents as functions of log t. To see this figure in color, go online.

Figure 2

(A) The mean-square displacement 〈Δx²(t)〉 of a particle along the axis of a tube with periodic spherical dead ends (inset). The system parameters in dimensionless units: tube and cavity radii are 0.8 and 1.5, respectively; intercavity distance is 2; radius of the apertures connecting the tube with the cavities is 0.1; and particle diffusion coefficient in free space is 1. (Dotted and dashed straight lines) Short- and long-time asymptotic behaviors of 〈Δx²(t)〉, respectively. (B) The three exponents α_fit(t), α_int(t), and α_dif(t) obtained by interpreting the transient behavior of 〈Δx²(t)〉 as anomalous subdiffusion. (Inset) The exponents as functions of log t. To see this figure in color, go online.