  We study subdiffusive overdamped Brownian ratchets periodically rocked by an
external zero-mean force in viscoelastic media within the framework of
non-Markovian Generalized Langevin equation (GLE) approach and associated
multi-dimensional Markovian embedding dynamics. Viscoelastic deformations of
the medium caused by the transport particle are modeled by a set of auxiliary
Brownian quasi-particles elastically coupled to the transport particle and
characterized by a hierarchy of relaxation times which obey a fractal scaling.
The most slowly relaxing deformations which cannot immediately follow to the
moving particle imprint long-range memory about its previous positions and
cause subdiffusion and anomalous transport on a sufficiently long time scale.
This anomalous behavior is combined with normal diffusion and transport on an
initial time scale of overdamped motion. Anomalously slow directed transport in
a periodic ratchet potential with broken space inversion symmetry emerges due
to a violation of the thermal detailed balance by a zero-mean periodic driving
and is optimized with frequency of driving, its amplitude, and temperature.
Such optimized anomalous transport can be low dispersive and characterized by a
large generalized Peclet number. Moreover, we show that overdamped subdiffusive
ratchets can sustain a substantial load and do a useful work. The corresponding
thermodynamic efficiency decays algebraically in time since the useful work
done against a load scales sublinearly with time following to the transport
particle position, but the energy pumped by an external force scales with time
linearly. Nevertheless, it can be transiently appreciably high and compare well
with the thermodynamical efficiency of the normal diffusion overdamped ratchets
on sufficiently long temporal and spatial scales.
