As an example of a hamiltonian ratchet in two-dimensional space but without driving, we study directed transport in a spatially periodic billiard immersed in a uniform stationary magnetic field. The magnetic field together with the geometry of the billiard are sufficient to break all symmetries relevant for transport, making an external driving unnecessary. In the high- and low-field limits, the motion becomes respectively pseudo-integrable and integrable; in both cases transport is absent. In the intermediate regime, surfaces of section reveal a mixed phase space, a necessary condition for directed currents. We find directed transport for individual invariant sets, while on average over the entire energy shell, a sum rule excludes finite transport. The dynamical mechanisms underlying directed transport are analyzed in terms of stable and unstable periodic orbits.
URL : http://ptp.ipap.jp/link?PTPS/150/313/
DOI : 10.1143/PTPS.150.313