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. In 1949 Fermi [1] proposed an acceleration mechanism of cosmic ray particles interacting with a time dependent magnetic field (for a review see [2]). Ever since, this has been a subject of intense study in a broad range of systems in various areas of physics, including astrophysics [3,4,5], plasma physics [6,7], atom optics [8,9] and has even been used for the interpretation of experimental results in atomic physics [10]. Furthermore, when the mechanism is linked to higher dimensional timedependent billiards, such as a time-dependent variant of the classic Lorentz Gas, it has profound implications on statistical and solid state physics [11]. Several modifications of the original model have been suggested, one of which is the well-known Fermi-Ulam model (FUM) [12,13,14] which describes the bouncing of a ball between an oscillating and a fixed wall. FUM and its variants have been the subject of extensive theoretical (see Ref.[13] and references therein) and experimental [15,16,17] studies as they are simple to conceive but hard to understand in that their behavior is quite complicated. A standard simplification [13] widely used in the literature, the static wall approximation (SWA), ignores the displacement of the moving wall but retains the time dependence in the momentum exchange between particle and wall at the instant of collision as if the wall were oscillating. The SWA speeds up time-consuming numerical simulations and allows semi-analytical treatments as well as a deeper understanding of the system [13,18,19,20,21]. However, as shown by Einstein in his treatment of the Brownian random walk [22], taking account of the full phase space trajectory (instead of the momentum component only) is essential for the correct description of diffusion processes. More recently, in the context of diffusion in the deterministic FUM, Lieberman et al have shown that one has to employ both canonical conjugate variables (position and momentum) in order to obtain the correct momentum distribution in the asymptotic steady state [20]. The present work shows that even in the absence of an asymptotic steady state the diffusion in velocity space is deeply affected by the location of th...
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.
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.
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.
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