Power-law frictional landscapes induce anomalous diffusion

Particle motion often exhibits anomalous diffusion arising from spatial inhomogeneity in the complex structures of soft materials. Spatial inhomogeneity induces a power-law frictional landscape for the mean-squared displacement of particles in a force-free environment; expressly, 〈x2(t)〉∼tα (i.e., 0<α≤2). We calculate analytically and numerically this mean-squared displacement. By comparing with a XY system in a logarithmic potential, for which anomalous diffusion transitions occur in regimes from normal diffusion to confinement, we investigated the diffusive dynamics of a realistic system away from its equilibrium state. Spatial nonlocal processes were found to be equivalent to time nonlocal ones (i.e., non-Ohmic memory); specifically, the weaker the effective friction produced, the stronger is the diffusion induced. This overcomes the difficulty encountered when evaluating memory effects in experiments. Aided by the generalized Green–Kubo formula, our model is also compared with diffusion processes obeying the scaling behavior of the velocity correlation function. The present study on anomalous diffusion in inhomogeneous environments is helpful because the phenomenon appears in soft, solid and biological matter.