Nonperiodic delay mechanism in time-dependent chaotic scattering

We study the occurence of delay mechanisms other than periodic orbits in systems with time dependent potentials that exhibit chaotic scattering. By using as model system two harmonically oscillating disks on a plane, we have found the existence of a mechanism not related to the periodic orbits of the system, that delays trajectories in the scattering region. This mechanism creates a fractal-like structure in the scattering functions and can possibly occur in several time-dependent scattering systems.Comment: 12 pages, 9 figure

A consistent approach for the treatment of Fermi acceleration in time-dependent billiards

The standard description of Fermi acceleration, developing in a class of time-dependent billiards, is given in terms of a diffusion process taking place in momentum space. Within this framework the evolution of the probability density function (PDF) of the magnitude of particle velocities as a function of the number of collisions $n$ is determined by the Fokker-Planck equation (FPE). In the literature the FPE is constructed by identifying the transport coefficients with the ensemble averages of the change of the magnitude of particle velocity and its square in the course of one collision. Although this treatment leads to the correct solution after a sufficiently large number of collisions has been reached, the transient part of the evolution of the PDF is not described. Moreover, in the case of the Fermi-Ulam model (FUM), if a stadanrd simplification is employed, the solution of the FPE is even inconsistent with the values of the transport coefficients used for its derivation. The goal of our work is to provide a self-consistent methodology for the treatment of Fermi acceleration in time-dependent billiards. The proposed approach obviates any assumptions for the continuity of the random process and the existence of the limits formally defining the transport coefficients of the FPE. Specifically, we suggest, instead of the calculation of ensemble averages, the derivation of the one-step transition probability function and the use of the Chapman-Kolmogorov forward equation. This approach is generic and can be applied to any time-dependent billiard for the treatment of Fermi-acceleration. As a first step, we apply this methodology to the FUM, being the archetype of time-dependent billiards to exhibit Fermi acceleration.Comment: 12 Pages, 7 figure

Quantum versus Classical Dynamics in a driven barrier: the role of kinematic effects

We study the dynamics of the classical and quantum mechanical scattering of a wave packet from an oscillating barrier. Our main focus is on the dependence of the transmission coefficient on the initial energy of the wave packet for a wide range of oscillation frequencies. The behavior of the quantum transmission coefficient is affected by tunneling phenomena, resonances and kinematic effects emanating from the time dependence of the potential. We show that when kinematic effects dominate (mainly in intermediate frequencies), classical mechanics provides very good approximation of quantum results. Moreover, in the frequency region of optimal agreement between classical and quantum transmission coefficient, the transmission threshold, i.e. the energy above which the transmission coefficient becomes larger than a specific small threshold value, is found to exhibit a minimum. We also consider the form of the transmitted wave packet and we find that for low values of the frequency the incoming classical and quantum wave packet can be split into a train of well separated coherent pulses, a phenomenon which can admit purely classical kinematic interpretation

A nonlinear classical model for the decay widths of Isoscalar Giant Monopole Resonances

The decay of the Isoscalar Giant Monopole Resonance (ISGMR) in nuclei is studied by means of a nonlinear classical model consisting of several noninteracting nucleons (particles) moving in a potential well with an oscillating nuclear surface (wall). The motion of the nuclear surface is described by means of a collective variable which appears explicitly in the Hamiltonian as an additional degree of freedom. The total energy of the system is therefore conserved. Although the particles do not directly interact with each other, their motions are indirectly coupled by means of their interaction with the moving nuclear surface. We consider as free parameters in this model the degree of collectivity and the fraction of nucleons that participate to the decay of the collective excitation. Specifically, we have calculated the decay width of the ISGMR in the spherical nuclei $^{208}\rm{Pb}$, $^{144}\rm{Sm}$, $^{116}\rm{Sn}$ and $^{90}\rm{Zr}$. Despite its simplicity and its purely classical nature, the model reproduces the trend of the experimental data which show that with increasing mass number the decay width decreases. Moreover the experimental results (with the exception of $^{90}\rm{Zr}$) can be well fitted using appropriate values for the free parameters mentioned above. It is also found that these values allow for a good description of the experimentally measured $^{112}\rm{Sn}$ and $^{124}\rm{Sn}$ decay widths. In addition, we give a prediction for the decay width of the exotic isotope $^{132}Sn$ for which there is experimental interest. The agreement of our results with the corresponding experimental data for medium-heavy nuclei is dictated by the underlying classical mechanics i.e. the behaviour of the maximum Lyapunov exponent as a function of the system size

Hyperacceleration in a stochastic Fermi-Ulam model

Fermi acceleration in a Fermi-Ulam model, consisting of an ensemble of particles bouncing between two, infinitely heavy, stochastically oscillating hard walls, is investigated. It is shown that the widely used approximation, neglecting the displacement of the walls (static wall approximation), leads to a systematic underestimation of particle acceleration. An improved approximative map is introduced, which takes into account the effect of the wall displacement, and in addition allows the analytical estimation of the long term behavior of the particle mean velocity as well as the corresponding probability distribution, in complete agreement with the numerical results of the exact dynamics. This effect accounting for the increased particle acceleration -Fermi hyperacceleration- is also present in higher dimensional systems, such as the driven Lorentz gas.Comment: 4 pages, 3 figures. To be published in Phys. Rev. Let

Scattering off an oscillating target: Basic mechanisms and their impact on cross sections

We investigate classical scattering off a harmonically oscillating target in two spatial dimensions. The shape of the scatterer is assumed to have a boundary which is locally convex at any point and does not support the presence of any periodic orbits in the corresponding dynamics. As a simple example we consider the scattering of a beam of non-interacting particles off a circular hard scatterer. The performed analysis is focused on experimentally accessible quantities, characterizing the system, like the differential cross sections in the outgoing angle and velocity. Despite the absence of periodic orbits and their manifolds in the dynamics, we show that the cross sections acquire rich and multiple structure when the velocity of the particles in the beam becomes of the same order of magnitude as the maximum velocity of the oscillating target. The underlying dynamical pattern is uniquely determined by the phase of the first collision between the beam particles and the scatterer and possesses a universal profile, dictated by the manifolds of the parabolic orbits, which can be understood both qualitatively as well as quantitatively in terms of scattering off a hard wall. We discuss also the inverse problem concerning the possibility to extract properties of the oscillating target from the differential cross sections.Comment: 18 page