Exponential Fermi acceleration in general time-dependent billiards

We show, that under very general conditions, a generic time-dependent billiard, for which a phase space of corresponding static (frozen) billiards is of the mixed type, exhibits the exponential Fermi acceleration in the adiabatic limit. The velocity dynamics in the adiabatic regime is represented as an integral over a path through the abstract space of invariant components of corresponding static billiards, where the paths are generated probabilistically in terms of transition-probability matrices. We study the statistical properties of possible paths and deduce the conditions for the exponential Fermi acceleration. The exponential Fermi acceleration and theoretical concepts presented in the paper are demonstrated numerically in four different time-dependent billiards.

Figure 1

Two $\zeta$ trajectories, namely ${\zeta }_a$ and ${\zeta }_b$, which are represented as straight vertical lines, for which $F_1\left({\zeta }_a\right)=a_1+a_2=0$ and $F_1\left({\zeta }_b\right)=b_1+b_2=0$, are mixing with the probability $P_{n,m}^k$ from the invariant component $n$, which is the state on the trajectory ${\zeta }_a$ at time $j=k$, to the invariant component $m$, which is the state of the trajectory ${\zeta }_b$ at time $j=k+1$. This allows an additional $\zeta$ trajectory for which $F_1=a_1+b_2\ne 0$ if $a_1\ne b_1$.

Figure 2

Numerical study of the phase space of a time-dependent billiard defined in (24) with the deformation parameter $a=0.3$ (a). [(b), (c), and (d)] Projections on the Poincare line of section $y=0$ [the dashed line in (a)]; gray denotes chaotic and white regular regions of the corresponding static billiards; in light gray, contours of constant $|f^{\prime }|$, Eq. pp2, on 11 equidistant levels between 0 and $f_{max}^{\prime }$. (b) Phase-space structures of four corresponding static billiards (upper-left). [(c) and (d)] Time evolution of two slices [dashed lines in (b)]. Expanding and contracting phases are symmetric: expanding phase = direction up, contracting phase = direction down. Lines with arrows are fractions of two trajectories in two different projections: a time of one period was divided into 200 subintervals on which the value of local minimum of $x$ (and $cos\theta$) of a trajectory was determined and used in the plot instead of all intersections with the surface of section. Parts of trajectories in the chaotic region are not plotted. Solid blue and dashed red represent the expanding and contracting phases, respectively. The velocities of considered trajectories are $\sim {10}^5$. Both trajectories start in the chaotic component at the beginning of the expanding phase, which is at the bottom of the diagrams. The trajectory 1 is a typical example for which $F_1>0$, as can be seen from the path through the contours of constant $|f^{\prime }|$, while the trajectory 2 is symmetric and thus $F_1\to 0$.

Figure 3

(a) A linear increase of the average velocity with respect to the number of collisions $n$ (exponential acceleration in the continuous time $t$); ${10}^3$ initial conditions used. (b) The distribution of $F_1$ for different $v_0$ and for ${10}^6$ initial conditions uniformly distributed in $\{\mathbf{r},\theta \}$ at $t=0$, when the billiard is almost completely chaotic. The central peak is growing as $\sqrt{v_0}$ and converges to the Dirac $\delta$ distribution. The peak corresponds to symmetric $\zeta$ trajectories which pass over the same sequence of invariant components in both the expanding and contracting phases. The distribution ${\rho }_{\pi }\left(F_1\right)$ is effectively independent of the velocity and has a positive mean and a finite width, which implies the exponential acceleration. (c) The same as (b) but with ${log}_{10}{\rho }_{\pi }\left(F_1\right)$ on the ordinate instead.

Figure 4

The acceleration for different deformation parameters $a$ in a billiard with the boundary as defined in (24). We see that smaller deformations (smaller $a$) correspond to a slower transition to the exponential acceleration (solid lines with slopes 1).

Figure 5

The acceleration of particles (${10}^3$ initial conditions at $v_0=1$) in time-dependent oval billiard with $\epsilon =0.4$ and deformation parameters ${\eta }_1=0$ and ${\eta }_2=0.1$ (a). The acceleration exponent $\beta$ is far from the expected asymptotic value $\beta =1$. The distribution of $F_1$ steel strongly depends on the velocity (b), satisfying the scaling property (15) for fully chaotic systems. However, we see that tails are already independent of the velocity, thus in the deep adiabatic limit the mechanism of exponential acceleration should prevail. The distributions in (b) are calculated with ensembles of ${10}^4$ initial conditions.

Figure 6

The acceleration of particles (${10}^3$ initial conditions at $v_0=1$) in a time-dependent oval billiard with $\epsilon =0.2$ and deformation parameters ${\eta }_1=0$ and ${\eta }_2=0.9$ (a). The acceleration exponent $\beta$ is approaching $\beta =1$. The distribution of $F_1$ is (effectively) independent of the velocity in the regime $v>{10}^3$ (b), which is the indicator of the exponential acceleration ($\beta =1$). The distributions in (b) are calculated with ensembles of ${10}^4$ initial conditions.

Figure 7

A trajectory in the time-dependent Robnik billiard, Eq. (27).

Figure 8

The acceleration of particles (${10}^3$ initial conditions at $v_0=1$) in a time-dependent Robnik billiard, Eq. (27). The distribution of $F_1$ is (effectively) independent of the velocity in the regime $v>{10}^4$, while at $v={10}^3$ the distribution is still noticeably wider (b). The exponential acceleration is clearly visible only after ${10}^8$ collisions (a).

Figure 9

A trajectory in the time-dependent annular billiard and the evident exponential acceleration.