We discuss a non-reversible Markov chain Monte Carlo (MCMC) algorithm for particle systems, in which the direction of motion evolves deterministically. This sequential direction-sweep MCMC generalizes the widely spread MCMC sweep methods for particle or spin indices. The sequential 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 simplified two-dimensional dipole model, we show rigorously that sequential MCMC leaves the stationary probability distribution unchanged, yet 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. We show that sequential MCMC can have shorter mixing times than the algorithms with random updates of directions. We point out possible applications of sequential MCMC in polymer physics and in molecular simulation.
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