Criticality and Strong Intermittency in the Lorentz Channel

We demonstrate the emergence of criticality due to power-law cross correlations in an ensemble of noninteracting particles propagating in an infinite Lorentz channel. The origin of these interparticle long-range correlations is the intermittent dynamics associated with the ballistic corridors in the single particle phase space. This behavior persists dynamically, even in the presence of external driving, provided that the billiard’s horizon becomes infinite at certain times. For the driven system, we show that Fermi acceleration permits the synchronization of the particle motion with the periodic appearance of the ballistic corridors. The particle ensemble then acquires characteristics of self-organization as the weight of the phase space regions leading to critical behavior increases with time.

Figure 1

Schematic of the billiard geometry. The central scatterers of radius $b$ are driven harmonically, while the scatterers with radius $a$ remain fixed. $\mathbf{A}$ and $\theta$ are the amplitude and angle of oscillation, respectively. The system is infinite in the $x$ direction, while in the $y$ direction two hard flat boundaries are placed at $\pm \sqrt{3}w/2$, confining the motion of the particles in this direction. The oscillating disks are depicted in their equilibrium positions (thick black circles) together with the turning point positions (blue lower left and red upper right circles).

Figure 2

(a) The probability $P(N)$ of crossing $N$ cells without changing direction of motion with $b=\sqrt{3}/2-a\approx 0.386$, $|\mathbf{A}|=1/2-b$ and $\theta =0$ [inset], $\theta \approx 0.11$ [red diagonal crosses], $\theta \approx 0.28$ [green open squares], and $\theta =0.5$ [blue diamonds]. The magnitude of the initial particle velocities is fixed to $|{\mathbf{V}}_0|=A/10$. (b) 2-D histogram of ($V$, $\xi$) pairs leading to collisionless ballistic flights crossing from 10 to 40 cells ($V=|\mathbf{V}|$ is the velocity magnitude of a trajectory and $\xi$ is the oscillation phase of the central disk). (c) As in (b), using ($V$, $\varphi$) pairs ($\varphi$ is the angle of the corresponding total velocity with the $x$ axis). (d) $\varphi$ and (e) the velocity component $V_x$ versus the discretized time variable $m$.

Figure 3

Contour plot of $\tilde{C}$ for 25 different values of $b$ and 10 values of $\theta$. The black area corresponds to the noncritical state [exponential $P(N)$]. (a) Synchronous oscillations of the central disks, (b) uniformly distributed random initial phases. We use $|\mathbf{A}|=1/2-b$.

Figure 4

(a) Cross correlation function $G_{\mathrm{DIH}}$ of independent trajectories in a DIH setup, for $b=0.16$. (b) $G_{\mathrm{FH}}(n)$ for a FH setup with $b=a=0.48$. (c) Conditional probability $P(N|N_c)$ for $N_c=1$ (black crosses), $N_c=2$ (blue triangles), and $N_c=3$ (red crosses). The solid red (dashed green) line has slope $-2$ ($-3$).