Vertical chaos and horizontal diffusion in the bouncing-ball billiard

The bouncing-ball billiard is a low-dimensional system with which transport properties of real physical systems can be studied theoretically. We study the bouncing-ball billiard with nonconvex scatterers and small slopes. We show that between the horizontal and vertical motion there is a separation of time scales, which is controlled by the slope of the billiard. We apply the theory of time-scale separation developed by Kantz Physica D 187, 200 (2004). If the vertical motion is chaotic, the horizontal motion is diffusive, but if the vertical motion is (quasi)periodic, there is no diffusion. We confirm the results with numerical simulations. Hence, the order-chaos transition in the vertical degrees of freedom translates into a localization-delocalization transition for the horizontal motion.