Subdiffusion in time-averaged, confined random walks

Certain techniques characterizing diffusive processes, such as single-particle tracking or molecular dynamics simulation, provide time averages rather than ensemble averages. Whereas the ensemble-averaged mean-squared displacement (MSD) of an unbounded continuous time random walk (CTRW) with a broad distribution of waiting times exhibits subdiffusion, the time-averaged MSD, δ², does not. We demonstrate that, in contrast to the unbounded CTRW, in which δ² is linear in the lag time Δ, the time-averaged MSD of the CTRW of a walker confined to a finite volume is sublinear in Δ, i.e., for long lag times δ²~Δ^1-α. The present results permit the application of CTRW to interpret time-averaged experimental quantities.