Propagators and time-dependent diffusion coefficients for anomalous diffusion

Complex diffusive dynamics are often observed when one is investigating the mobility of macromolecules in living cells and other complex environments, yet the underlying physical or chemical causes of anomalous diffusion are often not fully understood and are thus a topic of ongoing research interest. Theoretical models capturing anomalous dynamics are widely used to analyze mobility data from fluorescence correlation spectroscopy and other experimental measurements, yet there is significant confusion regarding these models because published versions are not entirely consistent and in some cases do not appear to satisfy the diffusion equation. Further confusion is introduced through variations in how fitting parameters are reported. A clear definition of fitting parameters and their physical significance is essential for accurate interpretation of experimental data and comparison of results from different studies acquired under varied experimental conditions. This article aims to clarify the physical meaning of the time-dependent diffusion coefficients associated with commonly used fitting models to facilitate their use for investigating the underlying causes of anomalous diffusion. We discuss a propagator for anomalous diffusion that captures the power law dependence of the mean-square displacement and can be shown to rigorously satisfy the extended diffusion equation provided one correctly defines the time-dependent diffusion coefficient. We also clarify explicitly the relation between the time-dependent diffusion coefficient and fitting parameters in fluorescence correlation spectroscopy.