Dispersion of particles in an infinite-horizon Lorentz gas

We consider a two-dimensional Lorentz gas with infinite horizon. This paradigmatic model consists of pointlike particles undergoing elastic collisions with fixed scatterers arranged on a periodic lattice. It was rigorously shown that when $t\to \infty$, the distribution of particles is Gaussian. However, the convergence to this limit is ultraslow, hence it is practically unattainable. Here, we obtain an analytical solution for the Lorentz gas' kinetics on physically relevant timescales, and find that the density in its far tails decays as a universal power law of exponent $-3$. We also show that the arrangement of scatterers is imprinted in the shape of the distribution.

Figure 1

(a) In the two-dimensional Lorentz gas, a particle is moving with a constant speed, elastically colliding with circular scatterers residing on a square lattice. (b) Due to the infinite corridors, the particles exhibit long flights along the axes. (c) These power-law distributed times of travel are responsible to the power-law decay of their respective probability density function, Eq. (1), resulting in a cumulative distribution function (CDF) of the form $\text{CDF}\left(\tau \right)={\int }_0^{\tau }d{\tau }^{\prime }\psi \left({\tau }^{\prime }\right)\simeq 1-{\tau }_0^2/{\tau }^2$ (solid line). The stairlike structure of the CDF (circles) originates from the discrete nature of the lattice of scatterers. We used $R=0.4$ as the radius of the scatterers, with lattice constant of $a=1$. The speed $V$ was chosen to be one.

Figure 2

The probability density functions of a numerical simulation of the Lorentz gas with two open horizons (a) and the Lévy walk (LW) theory (b) for duration $t={10}^4$. The non-Gaussian crosslike shape clearly illustrates the sensitivity of the spreading density to the underlying structure of the square lattice of scatterers. The LW approximation Eq. (5) is in agreement with the simulation without any fitting. Further details can be found in the Supplemental Material [27].

Figure 3

Cross sections of the Lorentz gas' probability density function (PDF) and the PDF given by Eq. (5) for $t={10}^4$. The theory matches the simulation perfectly, both in the direction of an infinite corridor parallel to the horizontal symmetry axis (a), $y=0$, as well as in the direction of the main diagonal (b), $y=x$. The dotted green line represents Bleher's limiting law which is valid at $t\to \infty$. For (a), a linear-scaled center part is given in the inset. As this is the infinite-horizon direction, we see a power-law decay. This is in contrast with (b), where we see a fast decay with $x$, more similar to a Gaussian, due to the diagonal being blocked by scattering centers. The deviation in the last two data points of (b) originates from the finite number of sampled trajectories $\approx {10}^9$. Further details can be found in the Supplemental Material [27].