Classical scattering from oscillating targets
Introduction
Complex phenomena arising due to the interplay between chaotic and regular motion of dynamical systems have been the subject of many investigations over the last decades 1. A variety of physical, chemical and biological systems, being of Hamiltonian or dissipative origin, have been studied in detail. In particular, the scattering dynamics of two degrees of freedom conservative Hamiltonian systems is well understood 2. Recently interest has arisen in systems whose dynamics define a higher-dimensional phase space. This situation is realized either with several (⩾3) degrees of freedom or for explicitly time-dependent Hamiltonian systems. The latter imply the nonconservation of the energy. Specifically, the interest in time-dependent systems is motivated by their experimental accessibility. Mesoscopic systems showing quantum and coherent features on an intermediate scale, i.e., between microscopic and macroscopic dimensions, can nowadays be prepared and controlled to a large extent. Examples are micro- or nanostructures driven by external voltages or applied laser fields 3.
Unstable periodic orbits (UPOs) play the key role in the understanding of complex dynamical behaviour. They form the skeleton of the chaotic phase space and determine both the classical as well as the quantum properties of a system. In the case of time-independent Hamiltonian scattering systems of two degrees of freedom, chaotic scattering or topological chaos occurs when the invariant manifolds of the UPOs, which are accessible from the asymptotic region, show homoclinic or heteroclinic connections [4], [5]. These connections imply that the scattering functions display a fractal set of singularities. These properties have been found in a variety of physical problems like scattering in billiards [6], [7], [8], potential scattering 9, the restricted three-body problem [10], [11] and many more (see Ref. 2). Another focus of interest has been time-dependent (kicked) systems with one degree of freedom [12], [13], [14]. The common feature of all of the above systems is the low dimensionality of their corresponding phase space which allows the use of standard tools for the analysis of the chaotic dynamics, such as Poincaré surfaces of section.
Much less is known about the scattering dynamics in systems with extended phase space (effective phase space dimension >3). Only very recently some studies have been performed in this direction: the driven Helium atom 15, time-dependent billiard systems [16], [17] and scattering in model potentials 18 are examples. In particular in Ref. 18 the “difficulties” to observe the typical features of chaotic scattering in systems with many degrees of freedom are addressed. A characteristic example of such a high-dimensional system is the scattering of a particle from two harmonically oscillating hard disks. In a recent work on this system 17 it was found that an infinite sequence of peaks appears in the escape time (from the scattering region) as a function of the incoming velocity of the scatterer. These peaks do not represent singularities but are related to an energy transfer process from the disk to the incoming particle leading to small outgoing velocities of the particle. These peaks accumulate for initial velocity v0→0. Since the system of two harmonically oscillating disks possesses a family of infinitely many UPOs, it is natural to pose the question whether the above-mentioned peak structure in the scattering functions is due to the manifolds of the UPOs. The mentioned accumulation tendency has also been observed in systems with attractive 1/r potentials and is understood in these systems in terms of the appearance of marginally stable periodic orbits, i.e., parabolic orbits [11], [15].
One purpose of this work is to find out what properties of the scattering functions are independent of the presence of the UPOs. To separate the influence of the UPOs we investigate a system which possesses no periodic orbits at all: the planar scattering from a single harmonically oscillating hard disk. Thus the system exhibits no topological chaos. The dynamics of the model takes place in a five-dimensional phase space. In spite of its simple set up, we show that the scattering functions have a rich structure. In particular, we study the behaviour of the escape time as a function of the velocity of the incoming particle, and demonstrate that several features can be observed. Examples are the infinite sequence of low-velocity peaks when the incoming particles are synchronized with the phase of the disk, and the existence of trajectories that display arbitrarily many bounces with the disk within a finite time. This interesting behaviour is further investigated by performing analytical calculations for a 1D-model.
The Letter is organized as follows. In Section 2 we describe the scattering system under investigation. In Section 3 we present numerical results on relevant scattering functions of the moving disk focusing on the low-velocity peaks. Motivated by the mechanism that results in low-velocity peaks, we study analytically the scattering off a wall with a simple time-law. Section 4 is devoted to study the existence of scattering events exhibiting a large number of bounces with the target. The mechanism involved is again put in a universal framework for oscillating targets. Finally, in Section 5 we provide a short summary and concluding remarks, as well as a brief discussion of a possible experiment.
Section snippets
Description of the scattering system
We study the scattering motion of a particle with initial velocity (vx0,vy0) at the point (x0,y0) of the plane, from a hard disk oscillating according to the law Here, is the position vector of the center of the disk with radius R, is a vector directed along the axis of oscillation of the disk with magnitude equal to the amplitude of the oscillation, and ω is the corresponding angular frequency. The initial phase φ0 of the disk determines the position of the disk at
Numerical results
We begin by considering the following scattering experiments. The particle is injected from the initial position (x0<−A−R) with the magnitude of the velocity and direction α towards the oscillating disk at time t=0, when the disk has a fixed initial phase φ0. In the investigation presented in this section, the velocity v0 of the particle is varied continuously, keeping the initial phase φ0 of the disk as well as the angle α fixed. In other words, we assume a synchronization
Multiple collision processes
In Fig. 2(c) we see that many collisions of the projectile with the oscillating target are possible. To obtain further insight into this property, we consider the following situation. Let us assume the incoming projectile is emitted with velocity v0 parallel to the x-axis (α=0) from the position (−A−R,0), which is the outermost left point of the collision region. We shall vary the initial velocity v0 of the projectile while keeping the initial phase φ0 of the target fixed. We repeat this
Concluding remarks
In this Letter we have investigated the scattering dynamics of a classical particle from a single oscillating target. The existence of a process that leads to a very large time of flight due to an energy loss mechanism has been shown. However, the time spent by the projectile in the collision region is always bounded, as expected from the non-existence of UPOs. We have also shown the existence of scattering events with infinitely many bounces. These properties are universal for oscillating
Acknowledgements
Financial support in the framework of the IKYDA program of the DAAD (Germany) and IKY (Greece) is gratefully acknowledged. L.B. acknowledges the kind hospitality of the Max-Planck-Institute for Nuclear Physics at Heidelberg. We thank Dr. E. Mavrommatis for helpful discussions.
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