Normal and Anomalous Diffusion in Soft Lorentz Gases

Motivated by electronic transport in graphenelike structures, we study the diffusion of a classical point particle in Fermi potentials situated on a triangular lattice. We call this system a soft Lorentz gas, as the hard disks in the conventional periodic Lorentz gas are replaced by soft repulsive scatterers. A thorough computational analysis yields both normal and anomalous (super)diffusion with an extreme sensitivity on model parameters. This is due to an intricate interplay between trapped and ballistic periodic orbits, whose existence is characterized by tonguelike structures in parameter space. These results hold even for small softness, showing that diffusion in the paradigmatic hard Lorentz gas is not robust for realistic potentials, where we find an entirely different type of diffusion.

Figure 1

The soft Lorentz gas: A point particle moves in a plane of partially overlapping Fermi potentials (inset) whose centers are situated on a triangular lattice (main figure). The dotted lines are contour lines, $w$ denotes the minimal distance between adjacent potentials for total energy $E=1/2$, and $A$ defines a triangular unit cell. The inset shows Fermi potentials along the dashed (blue) line in the main part for different values of the softness parameter $\sigma$ defined in Eq. (1).

Figure 2

Diffusion coefficient $D$ as a function of the gap size $w$. The (blue) wiggled line shows simulation results for $D(w)$ Eq. (2) in a slightly softened potential [$\sigma =0.05$ in Eq. (1)]. The thick (orange) line represents the corresponding analytical random walk approximation $D_B$, the thin (red) line the numerical $D_{\mathrm{B},\mathrm{num}}$ as explained in the text. The labeled numbers 3(a) to 3(d) refer to the periodic orbits depicted in Fig. 3. Gray columns indicate parameter intervals in which $D(w)$ does not exist. The inset displays $D(w)$ obtained from simulations for the conventional hard Lorentz gas [49].

Figure 3

Periodic orbits and islands of periodicity in phase space at different parameter values $w$ corresponding to Fig. 2. Shown in position space are characteristic periodic orbits for (a) $w=0.234$, (b) $w=0.31$, (c) $w=0.46$, (d) $w=0.18$. (a) and (b) feature quasiballistically propagating orbits yielding superdiffusive parameter regions in Fig. 2 while (c) and (d) generate local minima in $D(w)$. The insets display associated islands of periodicity in the Poincaré surface of section phase space $(x,\sin \theta )$ as defined in the text.

Figure 4

Regions of periodic orbits in the parameter space of gap size $w$ and potential softness parameter $\sigma$. Blue dots represent localized periodic orbits like (c) and (d) in Fig. 3, while red dots correspond to quasiballistic orbits like 3 and 3 therein. The black horizontal line at $\sigma =0.05$ yields a cut through the parameter space corresponding to the diffusion coefficient $D(w)$ in Fig. 2.