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...

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.

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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.

We consider the relation between relaxation time and the largest Lyapunov exponent in a system of two coupled oscillators, one of them being harmonic. It has been found that in a rather broad region of parameter space, contrary to the common expectation, both Lyapunov exponent and relaxation time increase as a function of the total energy. This effect is attributed to the fact that above a critical value of the total energy, although the Lyapunov exponent increases, Kolmogorov-Arnold-Moser tori appear and the chaotic fraction of phase space decreases. We examine the required conditions and demonstrate the key role of the dispersion relation for this behavior to occur. This study is useful, among other things, in the understanding of the damping of nuclear giant resonances.