  We investigate the transport and separation of overdamped particles under the
action of a uniform external force in a two-dimensional periodic energy
landscape. Exact results are obtained for the deterministic transport in a
square lattice of parabolic, repulsive centers that correspond to a
piecewise-continuous linear-force model. The trajectories are periodic and
commensurate with the obstacle lattice and exhibit phase-locking behavior in
that the particle moves at the same average migration angle for a range of
orientation of the external force. The migration angle as a function of the
orientation of the external force has a Devil's staircase structure. The first
transition in the migration angle was analyzed in terms of a Poincare map,
showing that it corresponds to a tangent bifurcation. Numerical results show
that the limiting behavior for impenetrable obstacles is equivalent to the high
Peclet number limit in the case of transport of particles in a periodic pattern
of solid obstacles. Finally, we show how separation occurs in these systems
depending on the properties of the particles.
