Rare events and their impact on velocity diffusion in a stochastic Fermi-Ulam model

A simplified version of the stochastic Fermi-Ulam model is investigated in order to elucidate the effect of a class of rare low-velocity events on the velocity diffusion process and consequently Fermi acceleration. The relative fraction of these events, for sufficiently large times, decreases monotonically with increasing variance of the magnitude of the particle velocity. However, a treatment of the diffusion problem which totally neglects these events, gives rise to a glaring inconsistency associated with the mean value of the magnitude of the velocity in the ensemble. We propose a general scheme for treating the diffusion process in velocity space, which succeeds in capturing the effect of the low-velocity events on the diffusion, providing a consistent description of the acceleration process. The present study exemplifies the influence of low-probability events on the transport properties of time-dependent billiards.

Figure 1

Numerical results (solid line) for $⟨\delta \mid {\tilde{V}}_n\mid ⟩=\frac{1}{n}{\sum }_{i=1}^n⟨\delta \mid V_i\mid ⟩$ on the basis of an ensemble of $5\times {10}^4$ particles evolving in a phase randomized FUM as a function of the number of collisions. Results were obtained by iterating the exact map given by Eqs. (1) for $3\times {10}^5$ number of collisions, with $ϵ=\frac{1}{15}$. It is noted that $⟨\delta \mid {\tilde{V}}_n\mid ⟩$ is measured in units of $\omega w$. The numerical results were fitted with a power-law model, $⟨\delta \mid {\tilde{V}}_n\mid ⟩\propto n^k$, for $n>{10}^5$ yielding $k=0.492\pm {10}^{-3}$ (dashed line).

Figure 2

Histograms (bars) of the particle absolute velocities for ${10}^4$ and ${10}^5$ collisions. Numerical results were obtained on the basis of an ensemble of $1.6\times {10}^5$ trajectories. The analytical solution given by Eq. (10) (black line) is also shown. $\mid V\mid$ is measured in units of $\omega w$.

Figure 3

Numerical results (×) for the statistical weight of the set of collision events $C_2$ obtained on the basis of an ensemble of $1.6\times {10}^5$ trajectories propagated for $3\times {10}^5$ collisions. Results were smoothed using a $2\times {10}^3$ point moving average filter to reduce statistical fluctuations. Fitting with the model given by Eq. (11) yields $a=0.3183\pm {10}^{-4}$ (solid line).

Figure 4

Numerical results (×) for the statistical weight of the set of collision events $C_2$. Numerical results were obtained on the basis of an ensemble of $4\times {10}^4$ trajectories propagated for ${10}^6$ collisions and with a Gaussian initial distribution of particle absolute velocities centered at $\mid V\mid =4.5$, with variance ${\sigma }^2=\frac{100}{225}$. Results were smoothed using a $2\times {10}^4$ point moving average filter. The analytical expression given by Eq. (11) with $a=0.3183$ is also plotted for the sake of comparison.

Figure 5

Numerical results (bars) for the PDF of the phase upon collisions for events which are elements of the set $C_2$. Numerical results were obtained on the basis of an ensemble of $3\times {10}^4$ collision events, on which the particle velocity was “artificially” reversed. The analytical estimation of the PDF is also shown (solid line). The phase is given in units of $\pi$.

Figure 6

Numerical results (bars) for the PDF of the phase upon all types of collisions. Numerical results were obtained on the basis of an ensemble of ${10}^6$ collision events. The uniform distribution is also plotted for the sake of comparison (solid line). The phase axis is normalized by division with $\pi$.

Figure 7

Numerical results (solid line) for the cumulative ensemble mean absolute velocity of the particles undergoing collisions which lead to an artificial velocity reversal. Numerical results were obtained on the basis of an ensemble of $1.6\times {10}^5$ trajectories propagated for $3\times {10}^5$ collisions. The best linear fit is also shown (dashed line). It is noted that ${⟨\mid {\tilde{V}}_{n-1}\mid ⟩}_2$ is measured in units of $\omega w$.

Figure 8

Numerical results (×) for the ensemble mean absolute velocity as a function of the number of collisions $n$. Numerical results were obtained on the basis of an ensemble of $1.6\times {10}^5$ trajectories propagated for $3\times {10}^5$ collisions. The semianalytical estimation of Eq. (21) is also plotted (solid line). It is noted that $⟨\mid V_n\mid ⟩$ is measured in units of $\omega w$.