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

Abstract

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.

Universal energy diffusion in a quivering billiard

“…An analogous static wall approximation can be constructed for higher dimensional billiards [19,26,29]. Like the quivering billiard, the static wall approximation eliminates the implicit equations for the time between collisions by holding the billiard boundary fixed.…”

“…An analogous hopping wall approximation for two dimensions is presented in [29]. Like the static wall approximation, the hopping wall approximation eliminates the implicit equations for the time between collisions, which eases numerical and analytical study.…”

“…[28,29], Karlis et al show that the stochastic static wall map and its analogue for the two-dimensional Lorentz gas give one half the asymptotic energy growth rate of the stochastic Fermi-Ulam map. This inconsistency exists even for small a, so Karlis et al conclude that (69) is not a valid approximation of (68).…”

“…The two models differ because the static wall approximation assumes u(t b ) to be a well behaved function. It is common practice to consider stochastic versions of the maps (68) and (69), where u(t b ) is replaced by u(t b + ζ) for some random variable ζ [3,15,19,26,28,29]. The stochastic case simulates the effects of external noise on the system and allows one to average over ζ when determining ensemble averages, which often facilitates analytical calculations.…”

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

“…[2][3][4][5] One of the most studied versions of the problem is the onedimensional Fermi-Ulam model (FUM). [6][7][8][9][10] The model consists basically of a classical particle confined and bouncing between two rigid walls, one of them is assumed to be fixed and the other one moves according to a periodic function. For such a system, it is known that the phase space, in the absence of dissipation, is mixed, in the sense that depending on the combinations of control parameters and initial conditions, Kolmogorov-Arnold-Moser (KAM) islands, invariant spanning curves and chaotic seas are all observed.…”