  We perform 2.5D hybrid simulations to investigate the origin and evolution of
relative drift speeds between protons and $\alpha$ particles in the
collisionless turbulent low-$\beta$ solar wind plasma. We study the generation
of differential streaming by wave-particle interactions and absorption of
turbulent wave spectra. Next we focus on the role of the relative drifts for
the turbulent heating and acceleration of ions in the collisionless fast solar
wind streams. The energy source is given by an initial broad-band spectrum of
parallel propagating Alfv\'en-cyclotron waves, which co-exists with the plasma
and is self-consistently coupled to the perpendicular ion bulk velocities. We
include the effect of a gradual solar wind expansion, which cools and
decelerates the minor ions. This paper for the first time considers the
combined effect of self-consistently initialized dispersive turbulent
Alfv\'enic spectra with differentially streaming protons and $\alpha$ particles
in the expanding solar wind outflows within a 2.5D hybrid simulation study. In
the non-expanding wind, we find a threshold value of the differential streaming
$V_{\alpha p} = 0.5 V_\mathrm{A}$, for which the relative drift speed remains
nearly steady. For ions, streaming below the threshold value, the waves act to
increase the magnitude of the relative drift speed. Ions, which stream faster
than the threshold value become subject to nonlinear streaming instability and
as the system evolves their bulk velocities decrease. We find that the solar
wind expansion strongly affects the relative drift speeds and significantly
slows down both ion species for all values of the relative drift speeds
considered in this study.
