Statistical properties of the transport coefficient for deterministic anomalous diffusion

Anomalous transport has lately attracted much attention. In
particular, anomalous subdiffusion is frequently observed for
molecular transport in cells and plasma membranes. Moreover, intensive
theoretical studies have revealed novel statistical behaviors of
subdiffusion such as aging and weak ergodicity breaking. In this
study, statistical properties of the transport coefficient for
deterministic anomalous diffusion from the viewpoint of infinite
ergodic theory has been investigated. We show that diffusion
coefficients are intrinsically random. In the case of subdiffusion,
the distribution function obeys a Mittag-Leffler distribution and the
time-averaged mean square displacement grows linearly, even though the
ensemble-averaged mean square displacement grows sublinearly. On the
other hand, in superdiffusion, the distribution function obeys another
distribution function called the generalized arcsine
distribution. These distributional limit theorems are shown using the
infinite ergodic theory. The findings provide a novel relation between
an infinite invariant measure and transport coefficients in
nonequilibrium statistical mechanics.