  We discuss a non-reversible, lifted Markov-chain Monte Carlo (MCMC) algorithm
for particle systems in which the direction of proposed displacements is
changed deterministically. This algorithm sweeps through directions analogously
to the popular MCMC sweep methods for particle or spin indices. Direction-sweep
MCMC can be applied to a wide range of original reversible or non-reversible
Markov chains, such as the Metropolis algorithm or the event-chain Monte Carlo
algorithm. For a single two-dimensional dipole, we consider direction-sweep
MCMC in the limit where restricted equilibrium is reached among the accessible
configurations before changing the direction. We show rigorously that
direction-sweep MCMC leaves the stationary probability distribution unchanged,
and that it profoundly modifies the Markov-chain trajectory. Long excursions,
with persistent rotation in one direction, alternate with long sequences of
rapid zigzags resulting in persistent rotation in the opposite direction in the
limit of small direction increments. The mapping to a Langevin equation then
yields the exact scaling of excursions while the zigzags are described through
a non-linear differential equation that is solved exactly. We show that the
direction-sweep algorithm can have shorter mixing times than the algorithms
with random updates of directions. We point out possible applications of
direction-sweep MCMC in polymer physics and in molecular simulation.
