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.

Figure 1

Numerical results for $V_{\mathrm{rms}}$ of an ensemble of ${10}^4$ particles evolving in a FUM with two oscillating walls as a function of the number of collisions. Results were obtained by iterating the exact (circles) as well as the corresponding static (diagonal crosses) and hopping wall (upright crosses) approximative maps. It is noted that $V_{\mathrm{rms}}$ is measured in units of $\omega w$. Analytical results according to the SWA averaged over the random phase (solid line) as well as the analytical prediction according to Eq. (7) (dash-dotted line) are also shown.

Figure 2

Numerically computed PDF for the magnitude of 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) $n=5\times {10}^5$. In each case the analytical result given by Eq. (8) using the HWA is also shown (solid line). Finally, (c) shows the numerically obtained evolution of $⟨|V|⟩$ as a function of $n$ using the exact dynamics (open circles) as well as the analytical approximation based on Eq. (8) (solid line).

Figure 3

Numerical results for $V_{\mathrm{rms}}$ of an ensemble of ${10}^4$ particles evolving in a triangular harmonically driven Lorentz gas with randomly chosen direction of oscillation on each collision, as a function of the number of collisions. Results were derived by iterating the exact maps (circles) as well as the corresponding static (diagonal crosses) and hopping (upright crosses) approximations. The parameters $|\mathbf{A}|=0.01$, $\omega =1$, $w=2.15$, and $V_0=\frac{1}{2.15}$ have been used in the numerical simulations, with $A$ denoting the magnitude of the amplitude of oscillation, $\omega$ the angular frequency, $w$ the spacing between the disk centers at equilibrium, and $V_0$ the initial particle velocity. Velocities are measured in units $\omega w$.