Figure 1

Poincaré surfaces of section for the chaotic Hamiltonian ratchet. (a) At low kicking strength $(K=0.1)$ the asymmetry in the system is already apparent. (b) At $K=2$ the system is in the globally chaotic regime; note the absence of any islands/tori. It is in this regime that our numerical simulations are performed. Each plot was calculated by starting 400 initial trajectories evenly spaced over a range of $x:[0,2\pi ]$ and $p:[-10,10]$ then kicking each trajectory 200 times.Reuse & Permissions

Figure 2

Figure illustrates differential energy absorption for particles with positive and negative momenta. An ensemble of particles (all with $p=0$ at initial time, $K=1.6,a=0.5$) is evolved, and $⟨p^2⟩$ is calculated separately, at each time, for particles with positive and negative momenta. The two upper curves [near the $(-)$ sign] show $⟨p^2⟩$ as a function of time for particles with negative momentum and two different values of $b$, the two lower curves [near the $(+)$ sign] show the corresponding curves for particles with positive momentum. We see that particles with negative momenta, for a certain time period absorb energy faster than those with negative momenta. Note also that the behavior becomes linear after a certain time.Reuse & Permissions

Figure 3

(a) Average classical kinetic energy $E(p_0,N=100)$ plotted as a function of initial momentum $p_0$, calculated numerically for an ensemble of ${10}^6$ particles with $K=14,a=1∕2,b=0.005$ time $t=100$. In (b) and (c) we have removed the momentum-independent and symmetrical $\cos npb$ contributions by plotting $E_{\mathrm{asymm}}(P_0,N)=\frac{1}{2}[E(p_0,N)-E(-p_0,N)]$. The energy spread after 20 kicks is shown in (b) and after 100 kicks in (c). The dashed curves, are obtained by Fourier analysis of the numerical results. For clarity, these curves have been shifted vertically.Reuse & Permissions

Figure 4

Contributions to classical asymmetric energy diffusion that arise from $\sin \left(npb\right)$ terms in the first order correction to the diffusion constant are shown as a function of $K$. Analytically predicted amplitudes are compared with numerical results for varying numbers of kicks to highlight the time scales involved. The amplitudes are scaled to 100 kicks for comparison, i.e., $A_n^{\prime }=A_n(K^2∕6)(100∕N)$. One can see in (a) that the numerical results for the $\sin \left(pb\right)$ term show excellent agreement with the analytical prediction for all $K$, suggesting that this term continues to influence the final current past 100 kicks. In (b) one sees good agreement for both 20 kicks and 40 kicks up to fairly high $K$, whereas after approximately $K=10$ the 100 kick curve begins to depart markedly from the analytical prediction. This suggests that for high kick strength the contribution to the final current from the $\sin \left(2pb\right)$ term has been damped by 100 kicks. This effect is even more noticeable for $\sin \left(4pb\right)$ as shown in (c). Good agreement between numerics and analytics exists only up to approximately $K=5$ with the 100 kick curve becoming heavily damped soon after. One can clearly see the 40 kick curve departing from the analytical result more quickly and completely than the 20 kick result. These plots show that for increasing kick strength, the time scale over which each term contributes to the final current changes.Reuse & Permissions

Figure 5

Classical ratchet saturation time, measured in kicks, versus system parameters $b$ (a) and $K^2$ (b). (a) The classical ratchet time is measured when the deviation of a 100 kick running average falls below 2% of its maximum value. A very good numerical agreement (solid line) is shown to a fit of $2\pi ∕K^2{b}^2$ (dashed line). The value of $K$ was fixed at 1.6. (b) using the same measurement technique for (a) the ratchet time is plotted against $K^2$ for a fixed $b=0.1$. Again a nonlinear curve fit of $2\pi ∕K^2{b}^2$, (dashed line) is compared to the numerical results (solid line). Each point on each graph is a result of a classical calculation of 500 000 trajectories run over 10 000 kicks.Reuse & Permissions

Figure 6

Comparison of numerical and analytical average current as a function of kicking strength $K$. In (a) the leading order analytical term (dashed line) given by Eq. (4.2) is shown with the numerical result (circles) for $b=0.01$ and $a=1∕2$. Note that the current reversals are accurately predicted, as is the general trend of the numerical curve. Panel (b) shows the contribution from the first term in Eq. (4.2), due to the $\sin x$ part of the potential. The final panel (c) shows the contribution from the second term, due to $\sin 2x$. Both curves are plotted with the numerical result.Reuse & Permissions

Figure 7

Variation of average classical and quantum currents as a function of kick strength $K$. Numerical values in brackets indicate the value of $\hslash$. In the quantum case, where $b=0.1$ for each curve, one notes that there is an improving fit with the classical $b=0.1$ curve for decreasing $\hslash$. At low $K$ the break time for $\hslash =0.5$, 0.25 is too short, and the system localizes before the maximum amount of classical asymmetry has been reached. As $K$ is increased, $t^{*}$ increases and the classical and quantum plots show good agreement once $t^{*}\approx t_r$. While both the quantum and classical curves share the same crossing, the classical curve for $b=0.1$ does not show the positive peak that features in both quantum graphs. This is due to the fact that the ratchet time is now too short to allow any appreciable build up of classical asymmetry (shown by the scaled ratchet time curve). The classical peak is recovered at smaller $b$ (not shown), and therefore increasing $t_r$ for a given value of $K$.Reuse & Permissions