Horizons and free-path distributions in quasiperiodic Lorentz gases

We study the structure of quasiperiodic Lorentz gases, i.e., particles bouncing elastically off fixed obstacles arranged in quasiperiodic lattices. By employing a construction to embed such structures into a higher-dimensional periodic hyperlattice, we give a simple and efficient algorithm for numerical simulation of the dynamics of these systems. This same construction shows that quasiperiodic Lorentz gases generically exhibit a regime with infinite horizon, that is, empty channels through which the particles move without colliding, when the obstacles are small enough; in this case, the distribution of free paths is asymptotically a power law with exponent $-3$, as expected from infinite-horizon periodic Lorentz gases. For the critical radius at which these channels disappear, however, a new regime with locally finite horizon arises, where this distribution has an unexpected exponent of $-5$, previously observed only in a Lorentz gas formed by superposing three incommensurable periodic lattices in the Boltzmann-Grad limit where the radius of the obstacles tends to zero.

Figure 1

Periodic and quasiperiodic arrays of scatters. (a) A channel in a square lattice of obstacles. (b) Different channels [thin red (strong gray) lines] in the Penrose Lorentz gas. At the bottom are shown trajectories in the direction of a horizontal channel. (c) Combining two Penrose Lorentz gases, the second rotated by an angle $\frac{\pi }{20}$, blocks the channels. At the bottom some trajectories are shown in a direction between two of the channels of the two Penrose arrays. (d) Quasiperiodic array that results from projecting a three-dimensional (3D) lattice into a 2D subspace. At the bottom we show a channel and some trajectories in the direction of the channel.

Figure 2

(a) Projection method used to produce the Fibonacci chain. $E$ is the physical space, where the particles are projected to produce the Fibonacci chain. (b) Periodization of the Fibonacci chain; particles can move in only two directions.

Figure 3

Trajectory of a particle constrained to move in plane parallel to one of the faces of the cube, in the case that the plane $E$ is totally irrational. The trajectory of the particle inside a channel represented in (a) fills densely the face of the cube, represented in (1) as a gray background with part of the trajectory highlighted. If the channels become blocked (b), the trajectory can still be dense in the face of the cube (3) or becomes finite if the particles moves in a plane parallel to the face of the cube (2), since it collides with the part of the ellipse (shown in black) that is the intersection of the cylinder with this plane. The thick (red) arrows on (b) show the intersection between the planes. The bottom arrow shows only a point while the other shows a larger area of intersection.

Figure 4

Trajectory of a particle constrained to move in a plane parallel to one of the faces of the cube, in the case that the plane $E$ is totally rational: (a) in the presence of a channel of width $\mathrm{\Delta }$; (b) when the channels are blocked. Below are shown trajectories of particles moving in a face of the cube. (1) A particle in a channel; (2) and (3) show two cases where the particle touches tangentially the obstacle; (4) a particle that collides with an obstacle. The cases (3) and (4) are contained in the plane marked in a dark color in (b), while (4) is contained in one of the faces of the cube.

Figure 5

(a) Free path length distribution for radii $r=0.03$ [red (upper) line]; $r=0.30$ [green (light gray)]; and $r=0.36$ [blue (bottom)]. (b) The corresponding cumulative distribution functions with the same color (gray scale) code.

Figure 6

(a) Sketch of the geometry to calculate the free-path length distribution inside a channel (as in the periodic Lorentz gases). $\epsilon$ measures the distance of the particle from the edge of the channel; $\mathrm{\Delta }$ is the width of the channel. (b) and (c) show how $s$ is defined. (d) Another view of the periodized model of (b), showing the trajectory of two particles, one with velocity parallel to the direction of the channel (in black), an the other slightly deviated [in red (gray)]. This trajectories are also shown in (e), where is sketched the geometry to calculate the free-path length distribution in the locally finite horizon regime for a 2D quasiperiodic Lorentz gas.

Figure 7

Measurement of the maximum free-path length as a function of the radius of an obstacle for fixed direction (slope) in a 2D Lorentz gas with the square arrangement. The employed slopes are: $\frac{1}{\varphi \sqrt{\varphi +1}},\sqrt{2},\pi$ and the Liouville constant $\sim 0.110001000000000000000001$.