  Stationary states of random walks with finite induced drift velocity on
one-dimensional lattices with aperiodic disorder are investigated by scaling
analysis. Three aperiodic sequences, the Thue-Morse (TM), the paperfolding
(PF), and the Rudin-Shapiro (RS) sequences, are used to construct the aperiodic
disorder. These are binary sequences, composed of two symbols A and B, and the
ratio of the number of As to that of Bs converges to unity in the infinite
sequence length limit, but their effects on diffusional behavior are different.
For the TM model, the stationary distribution is extended, as in the case
without current, and the drift velocity is independent of the system size. For
the PF model and the RS model, as the system size increases, the hierarchical
and fractal structure and the localized structure, respectively, are broken by
a finite current and changed to an extended distribution if the system size
becomes larger than a certain threshold value. Correspondingly, the drift
velocity is saturated in a large system while in a small system it decreases as
the system size increases.
