Phase-space composition of driven elliptical billiards and its impact on Fermi acceleration

We demonstrated very recently [Lenz et al., New J. Phys. 11, 083035 (2009)] that an ensemble of particles in the driven elliptical billiard shows a surprising crossover from subdiffusion to normal diffusion in momentum space. This crossover is not parameter induced, but rather occurs dynamically in the evolution of the ensemble. In this work, we consider three different driving modes of the elliptical billiard and perform a comprehensive analysis of the corresponding four-dimensional phase space. The composition of this phase space is different in the high-velocity regime compared to the low-velocity regime. We will show, among others, by investigating periodic orbits and probability distributions of laminar phases that the stickiness properties, which eventually determine the diffusion, are intimately connected with this change in the composition of the phase space with respect to velocity. In the course of the evolution, the accelerating ensemble thus explores regions of varying stickiness, leading to the mentioned crossover in the diffusion.

Figure 1

(Color online) Poincaré surface of section of the static elliptical billiard $a=2,b=1$ (upper part). Rotators have an inner confocal elliptic caustic and librators generate two confocal hyperbolas as caustics.

Figure 2

(Color online) Regions in $F$ space that allow for the existence of periodic orbits as a function of the period. The existence of periodic orbits deep in the librator region, i.e., values of $F$ toward $-3$, requires comparatively large periods. Consequently, periodic orbits with short periods are located around the separatrix $\left(F=0\right)$.

Figure 3

(Color online) Different driving modes of the ellipse in coordinate space for three different phases $\xi$: (a) breathing mode, (b) quadrupole mode, and (c) constant eccentricity mode.

Figure 4

(Color online) Poincaré surface of section of the frozen billiard corresponding to the quadrupole mode at two different phases: (a) ${\xi }_0=0.2$ and (b) ${\xi }_0=\pi /2$. The fraction of librators (blue circular shaped curves) is much higher in (b) than in (a).

Figure 5

(Color online) Collision resolved phase-space density ${\rho }_{n_1,n_2}\left(F,\xi \right)$ (logarithmic color map): (a) $n_1=10,n_2={10}^2$ and (b) $n_1=4\times {10}^6,n_2={10}^7$. ${\rho }_{n_1,n_2}\left(F,\xi \right)$ is largest around $F\approx -2.5$ for small $n$ (a), whereas the $F\left(\varphi ,\alpha \right)\approx 0$ region is predominantly populated for high collision numbers (b). In the insets, the corresponding distributions ${\rho }_{n_1,n_2}\left(\varphi ,\alpha \right)$ are shown. (Reproduced from Ref. [25].)

Figure 6

(Color online) Velocity resolved phase-space density ${\rho }_{v_1,v_2}\left(F,\xi \right)$ (logarithmic color map). (a) For low $\left(0<v<1\right)$ velocities, the ensemble covers the whole accessible $F\times \xi$ plane; (b) for higher $\left(30<v<40\right)$ velocities, it is exclusively located around the $F\left(\varphi ,\alpha \right)\approx 0$ region. The insets show the corresponding ${\rho }_{v_1,v_2}\left(\varphi ,\alpha \right)$. (Reproduced from Ref. [25].)

Figure 7

Lyapunov exponents $\sigma$ for particles starting with $v_0=30$ and different initial $F_0$’s. Particles with $\sigma =0$ move on invariant KAM tori, whereas particles with $\sigma >0$ perform chaotic motion, either inside bounded (especially with respect to $v$) chaotic layers or inside chaotic channels which are unbounded in $v$.

Figure 8

(Color online) Poincaré surface of section of two infinite heavy walls oscillating with opposite phase (variant of the FUM). The first invariant spanning curve is shown as a thick solid line (in blue). In Fig. 9, the phase-space density $\rho \left(\varphi ,\alpha \right)$ of the driven elliptical billiard is shown for different values of $v$: these four values of $v$ are marked with dashed lines (in red).

Figure 9

(Color online) Velocity resolved phase-space density ${\rho }_{v_1,v_2}\left(\varphi ,\alpha \right)$ (logarithmic color map). The density is highest around $\varphi =\pi /2,3\pi /2,\alpha =\pi /2$ in (a); a weak dip appears in (b) around this position and gets more pronounced in (c) until ${\rho }_{v_1,v_2}\left(\varphi \approx \pi /2,3\pi /2,\alpha \approx \pi /2\right)=0$ in (d). The different values of $v$ correspond to velocities right (a) below and (b) above the minimum velocity and right (c) below and (d) above the maximum velocity of the FISC of the corresponding FUM (see Fig. 8).

Figure 10

(Color online) Boundaries of the invariant structures in $F$ space as a function of the velocity $v$. The chaotic sea (the area between the two curves) is extended over the whole available $F$ range at low $v$ and gets more and more squeezed into a thin channel with increasing $v$.

Figure 11

(Color online) Phase-space density ${\rho }_{v_1,v_2}\left(\varphi ,\alpha \right)$ for $v_1=10,v_2=13$ in the quadrupole mode (logarithmic scale color map, $C=0.1$). The black regions with $\rho \left(\varphi ,\alpha \right)=0$ correspond to inaccessible invariant structures emanating from whispering-gallery orbits.

Figure 12

(Color online) Density of periodic orbits up to period 100 as a function of $F$ with $v_1<v<v_2$ and $v_1=0,v_2=1$ in (a) and $v_1=30,v_2=40$ in (b), where $v$ is the mean velocity of a periodic orbit with period $p$, i.e., $v=1/p{\sum }_{i=1}^p{v}_i$. The $F\left(\varphi ,\alpha \right)\approx 0$ region is depleted for (a) low velocities, and is populated predominantly for (b) higher velocities. Exemplary, the distribution of periodic orbits is shown in the insets in the $\varphi \times \alpha$ plane, where the black curves are the $F=\text{const}$ isolines for $\xi =0$. (Reproduced from Ref. [25].)

Figure 13

(Color online) Density $\rho \left(v\right)$ of periodic orbits up to period 100 for the breathing mode $\left(C=0.1\right)$; note the logarithmic scale. The density is highest for low velocities $\left(0<v<3\right)$ and is suppressed significantly for high velocities $\left(30<v<40\right)$.

Figure 14

(Color online) Histogram of the real eigenvalues of the loxodromic and hyperbolic periodic orbits; note the pronounced peak at 1 (binning of ${10}^{-2}$ is used). The inset shows a magnification of the major peak at 1 (binning of $2\times {10}^{-7}$).

Figure 15

Location of periodic orbits up to period 50 in the quadrupole mode for $10<v<13$ $\left(C=0.1\right)$. Note that the region covered with periodic orbits is the same as the one where the phase-space density follows ${\rho }_{v_1=10,v_2=13}\left(\varphi ,\alpha \right)\ne 0$ (see Fig. 11).

Figure 16

(Color online) Probability distribution $P\left(l\right)$ (cumulative distribution) that a laminar phase possesses a length greater than or equal to $l$, for two different velocity regimes (breathing mode, $C=0.1$).

Figure 17

(Color online) The fraction of particles $f_{\text{l}am}\left(n\right)$ that are in a laminar phase (i.e., they are sticky) as a function of the number of collisions for the breathing and constant eccentricity modes $\left(C=0.1\right)$.

Figure 18

(Color online) The ensemble averaged velocity $⟨v⟩$ as a function of the number of collisions for different driving modes and two different amplitudes $C$ is shown: (a) breathing mode, (b) constant eccentricity mode, and (c) quadrupole mode; note the different $y$ scale in (b). In (a) and (c), asymptotically $v\sim n^{1/2}$ (normal diffusion), whereas in (b) $v\sim n^{\gamma }$ with $\gamma <1/2$ (subdiffusion). The power-law exponents $\gamma$ are specified in Table .