Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards

We study diffusion on a periodic billiard table with an infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a Lévy walk combining exponentially distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients, which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards [J. Machta and R. Zwanzig, Phys. Rev. Lett. 50, 1959 (1983)].

Figure 1

Infinite-horizon periodic billiard table of square cells of sides $\ell$. The disks centered at the cells' corners have radii $\rho =0.45\ell$. Infinite corridors of widths $\delta =\ell -2\rho$ span along horizontal and vertical axes, through the centers of each cell. The broken red line shows a trajectory.

Figure 2

Graph showing two-dimensional phase-space sets according to the distance separating phase points at the boundary between two cells from their next collision on a disk. The darkest color encodes sets of points whose next collision takes place in the cell they move into; the brighter the colors, the more distant the next collision. The two axes correspond, respectively, to $p_{\parallel }/p_0$, which denotes the fraction of the velocity component directed along the boundary; $-1<p_{\parallel }/p_0<1$ (here restricted to $-\delta /\ell <p_{\parallel }/p_0<\delta /\ell$); and the position $s$ along the boundary $-\delta /2<s<\delta /2$, where $\delta =\ell -2\rho \left(\rho =0.45\ell \right)$ is the width of the corridor. With these coordinates, the area corresponds to the natural invariant measure, which is normalized if divided by $2\delta$.

Figure 3

Computed slopes $\beta$ (blue, lower curve) and intercepts $\alpha$ (red, upper curve) of the mean squared displacement of point particles on the infinite-horizon Lorentz gas, fitted according to Eq. (9) for different values of the parameter $\delta$. The values found are here normalized by those predicted by Eqs. (10) and (11). To illustrate the fitting procedure, the inset shows the mean squared displacement (blue curve, including error bars) measured, as a function of time, for $\delta =0.03$. The red dashed line is the result of a linear fit performed in the interval marked by the two vertical lines; see Ref. [29] for further details on this procedure. The units are chosen such that $\ell \equiv p_0\equiv 1$.