Ratchetlike pulse controlling the Fermi deceleration and hyperacceleration

Abstract

Many years ago Enrico Fermi [1] suggested an acceleration mechanism of cosmic ray particles interacting with a time dependent magnetic field. Later on different versions of the original model for the Fermi acceleration were proposed. In the first one, the Fermi-Ulam (FU) model, a bouncing particle moves between a fixed surface and a parallel oscillating surface [2]. This model was shown to be chaotic [3, 4]. In order to improve simulations a simplified version of the FU model was proposed [3], called the static wall model. It ignores the displacement of the moving wall but keeps the essential information for the momentum transfer as the wall was oscillating. This static model was discussed in many aspects [3, 5, 6, 7], even for circular billiards [8]. Usually invariant curves in the phase space, found for higher velocities, prevent the particle to increase its kinetic energy without bounds. Recently the hopping wall approximation was proposed [9, 10] which takes into account the effect of the wall displacement, and allows the analytical estimation of the particle mean velocity. Compared to the simplified static model, the particle acceleration is enhanced. The second kind of Fermi accelerated model was proposed in 1977 by Pustyl’nikov [11], who considered a particle on a periodically oscillating horizontal surface in the presence of a gravitational field. The above topics got attention in various areas of physics, ranging from nonlinear physics [3, 4, 5, 6, 7, 8, 9, 10], atom optics [12, 13, 14], plasma physics [15, 16] to astrophysics [17, 18, 19]. In this paper we use an ac driven asymmetric pulse to control the acceleration (deceleration) in the simplified FU model. The pulse is a deformed sawtooth driving law for the moving wall. This Ratchetlike pulse differs from the ac driven asymmetric pulses (symmetric sawtooth) used for the Fermi acceleration in the early work of Lichtenberg et al. [3] and proposed recently to control the motion of magnetic flux quanta [20] and to analyze the relative efficiency of mechanism leading to increased acceleration in the hopping wall approximation [10]. In the simplified Fermi model [3] the particle is free to move between the elastic impacts with the walls. Consider that the moving wall oscillates between two extrema with amplitude v0. The gravitational force is considered zero. The system is described by a map M1(2)(Vn,�n) = (Vn+1,�n+1) which gives, respectively, the velocity of the particle, and the phase of the moving wall, immediately after the particle suffers a collision with the wall. Considering dimensionless variables the FU map with the deformed sawtooth wall is written as