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.

Figure 1

Numerical results for $V_{\mathrm{rms}}$ of an ensemble of ${10}^4$ particles evolving in a stochastic FUM with two oscillating walls as a function of the number of collisions. Results were obtained by iterating the exact $(\times )$ as well as the corresponding static $(+)$ approximate map, with $ϵ=1∕15$ and $V_0=100∕15$. $V_{\mathrm{rms}}$ is measured in units of $\omega w$. The analytical result according to the SWA, obtained by averaging over the random phase (solid line), is also shown.

Figure 2

Numerical results for $V_{\mathrm{rms}}$ of an ensemble of ${10}^4$ particles evolving in a stochastic FUM with two oscillating walls as a function of the number of collisions. Results were obtained by iterating the exact $(엯)$ as well as the corresponding HWA map $(\nabla )$, with $ϵ=1∕15$ and $V_0=100∕15$. It is noted that $V_{\mathrm{rms}}$ is measured in units of $\omega w$. The analytical result according to the HWA, Eq. (14), obtained by averaging over the random phase (solid line), is also shown.

Figure 3

The position and the velocity of one of the oscillating walls as a function of time. The straight line represents the trajectory of an incident particle.

Figure 4

The oscillation law for the FUM (with one oscillating wall). The borderlines of the zones discussed in the text are delimited by the dashed lines of slope $\lambda =\mid V_n\mid T∕A$. The zones are also marked by double arrows.

Figure 5

Numerical results $(\times )$ as well as analytical results [see Eq. (25)]—solid line—for $R_{h,L}\left(n\right)$ in the original FUM, employing sawtooth driving for $u_0=0.01$, $a=5\times {10}^{-3}$, and $b=0.67$.

Figure 6

Numerically computed PDF for the magnitude of the particle velocity using an ensemble of ${10}^4$ trajectories and following the exact dynamics of Eq. (1) for (a) $n=5\times {10}^4$ and (b) $n=5\times {10}^5$ with $ϵ=1∕15$ and $V_0=100∕15$. In each case, the analytical result given by Eq. (31) using the HWA is also shown (solid line).

Figure 7

Numerical results obtained by the iteration of the exact $(엯)$ map for the evolution of the ensemble mean magnitude of particle velocity $⟨\mid V\mid ⟩$ versus the number of collisions, $n$ for $ϵ=1∕15$ and $\mid V_0\mid =100∕15$. It is noted that $⟨\mid V\mid ⟩$ is measured in units of $\omega w$. The analytical result according to the HWA, Eq. (32) (solid line), is also shown.

Figure 8

The variables associated with an elementary collision process.

Figure 9

Probability distribution functions (PDFs) of the (a) incidence angle $a$ and (b) azimuthal angle $\varphi$. The theoretical predictions for the corresponding PDFs are also plotted (solid line).

Figure 10

Numerical results for the root mean square particle velocity $V_{\mathrm{rms}}$ as a function of the number of collisions $n$ obtained by the iteration of the exact map, Eq. (34) $(엯)$ (with $ϵ=1∕215$ and $V_0=100∕215$) for the driven Lorentz gas consisting of scatterers oscillating horizontally, with the oscillation phase shifted randomly when an incident particle leaves the elementary cell of the lattice. Corresponding results obtained utilizing the hopping $(+)$ and static $(\times )$ approximative maps are also presented. The curves obtained by fitting with the model $\sqrt{\mu {ϵ}^{2}n+V_0^2}$ are also plotted (dashed lines).

Figure 11

Numerical results for the root mean square particle velocity, $V_{\mathrm{rms}}$, as a function of the number of collisions $n$ obtained by the iteration of the exact map $(엯)$ (with $ϵ=1∕215$ and $V_0=100∕215$) for the setup with scatterers oscillating in a randomly chosen direction. Corresponding results obtained utilizing the hopping $(+)$ and static $(\times )$ approximative maps are also presented. The curves acquired by fitting with the model $\sqrt{\mu {ϵ}^{2}n+V_0^2}$ are also plotted (dashed lines).

Figure 12

Numerical results for the phase autocorrelation function $C\left(n\right)$ vs the number of collisions $n$ for the Lorentz gas consisting of scatterers oscillating in a randomly chosen direction.

Figure 13

Root mean square particle velocity, $V_{\mathrm{rms}}$, as a function of the number of collisions $n$ obtained by the iteration of the exact map $(엯)$ (with $ϵ=1∕215$ and $V_0=100∕215$) for the case of scatterers with randomized phase of oscillation. For the sake of comparison, the analytical result of Eq. (51) is also shown (solid line).

Figure 14

Numerically computed PDF for the magnitude of the particle velocity using an ensemble of ${10}^4$ trajectories and following the exact dynamics for: (a) $n=5\times {10}^4$ and (b) $n=5\times {10}^5$ with $ϵ=1∕215$ and $\parallel V_0\parallel =100∕215$. In each case, the analytical result provided by Eq. (58) is also shown (solid line).