Criticality and strong intermittency in the Lorentz channel

We demonstrate the emergence of criticality due to power-law cross correlations in an ensemble of noninteracting particles propagating in an infinite Lorentz channel. The origin of these interparticle long-range correlations is the intermittent dynamics associated with the ballistic corridors in the single particle phase space. This behavior persists dynamically, even in the presence of external driving, provided that the billiard's horizon becomes infinite at certain times. For the driven system, we show that Fermi acceleration permits the synchronization of the particle motion with the periodic appearance of the ballistic corridors. The particle ensemble then acquires characteristics of self-organization as the weight of the phase space regions leading to critical behavior increases with time.

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 have 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 standard 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.

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 noninteracting 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.

Rare events and their impact on velocity diffusion in a stochastic Fermi-Ulam model

A simplified version of the stochastic Fermi-Ulam model is investigated in order to elucidate the effect of a class of rare low-velocity events on the velocity diffusion process and consequently Fermi acceleration. The relative fraction of these events, for sufficiently large times, decreases monotonically with increasing variance of the magnitude of the particle velocity. However, a treatment of the diffusion problem which totally neglects these events, gives rise to a glaring inconsistency associated with the mean value of the magnitude of the velocity in the ensemble. We propose a general scheme for treating the diffusion process in velocity space, which succeeds in capturing the effect of the low-velocity events on the diffusion, providing a consistent description of the acceleration process. The present study exemplifies the influence of low-probability events on the transport properties of time-dependent billiards.

Fermi acceleration in the randomized driven Lorentz gas and the Fermi-Ulam model

Fermi acceleration of an ensemble of noninteracting particles evolving in a stochastic two-moving wall variant of the Fermi-Ulam model (FUM) and the phase randomized harmonically driven periodic Lorentz gas is investigated. As shown in [A. K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis, and P. Schmelcher, Phys. Rev. Lett. 97, 194102 (2006)], the static wall approximation, which ignores scatterer displacement upon collision, leads to a substantial underestimation of the mean energy gain per collision. In this paper, we clarify the mechanism leading to the increased acceleration. Furthermore, the recently introduced hopping wall approximation is generalized for application in the randomized driven Lorentz gas. Utilizing the hopping approximation the asymptotic probability distribution function of the particle velocity is derived. Moreover, it is shown that, for harmonic driving, scatterer displacement upon collision increases the acceleration in both the driven Lorentz gas and the FUM by the same amount. On the other hand, the investigation of a randomized FUM, comprising one fixed and one moving wall driven by a sawtooth force function, reveals that the presence of a particular asymmetry of the driving function leads to an increase of acceleration that is different from that gained when symmetrical force functions are considered, for all finite number of collisions. This fact helps open up the prospect of designing accelerator devices by combining driving laws with specific symmetries to acquire a desired acceleration behavior for the ensemble of particles.

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.

Polymer gel water equivalence and relative energy response with emphasis on low photon energy dosimetry in brachytherapy

The water equivalence and stable relative energy response of polymer gel dosimeters are usually taken for granted in the relatively high x-ray energy range of external beam radiotherapy based on qualitative indices such as mass and electron density and effective atomic number. However, these favourable dosimetric characteristics are questionable in the energy range of interest to brachytherapy especially in the case of lower energy photon sources such as 103Pd and 125I that are currently utilized. In this work, six representative polymer gel formulations as well as the most commonly used experimental set-up of a LiF TLD detector-solid water phantom are discussed on the basis of mass attenuation and energy absorption coefficients calculated in the energy range of 10 keV-10 MeV with regard to their water equivalence as a phantom and detector material. The discussion is also supported by Monte Carlo simulation results. It is found that water equivalence of polymer gel dosimeters is sustained for photon energies down to about 60 keV and no corrections are needed for polymer gel dosimetry of 169Yb or 192Ir sources. For 125I and 103Pd sources, however, a correction that is source-distance dependent is required. Appropriate Monte Carlo results show that at the dosimetric reference distance of 1 cm from a source, these corrections are of the order of 3% for 125I and 2% for 103Pd. These have to be compared with corresponding corrections of up to 35% for 125I and 103Pd and up to 15% even for the 169Yb energies for the experimental set-up of the LiF TLD detector-solid water phantom.