Normal diffusion in crystal structures and higher-dimensional billiard models with gaps

We show, both heuristically and numerically, that three-dimensional periodic Lorentz gases—clouds of particles scattering off crystalline arrays of hard spheres—often exhibit normal diffusion, even when there are gaps through which particles can travel without ever colliding—i.e., when the system has an infinite horizon. This is the case provided that these gaps are not “too large,” as measured by their dimension. The results are illustrated with simulations of a simple three-dimensional model having different types of diffusive regime and are then extended to higher-dimensional billiard models, which include hard-sphere fluids.

Figure 1

(Color online) Spherical scatterers (light color, purple online) and gaps (dark color, green online) in the 3D periodic Lorentz model discussed in the text: (a) vertical planar gaps for $a=0.25$ and $b=0.15$ and (b) vertical cylindrical gaps for $a=0.4$ and $b=0.4$ (a body-centered-cubic structure). The gaps are shown in a single unit cell, but in fact extend throughout the whole of space. (c) When $a=0.55$ and $b=0.4$, the scatterers overlap, leaving an infinite, connected available space for the particles, which is depicted; for clarity, the spheres are omitted. In this case, the horizon is finite—there are no gaps in the structure.

Figure 2

(Color online) Linear-log plot of $⟨r{\left(t\right)}^2⟩∕t$ vs $t$ in different diffusive regimes: finite horizon ($a=0.55$, $b=0.4$), cylindrical gaps in a body-centered-cubic lattice $(a=b=0.4)$, single large cylindrical gap ($a=0.4$, $b=0.21$), and simple cubic lattice ($a=0.4$, $b=0.0$) with planar gaps. Means are taken over up to $4\times {10}^7$ initial conditions; error bars are of the order of the symbol size. Linear growth (weak superdiffusion) occurs only when there are planar gaps.

Figure 3

(Color online) Growth rate $\gamma \left(q\right)$ of higher moments $⟨r^q\left(t\right)⟩$ as a function of $q$; geometries are as in Fig. 2. For $a=0.4$ and $b=0.21$, the means were calculated over $2.4\times {10}^8$ initial conditions, up to a time $t=10000$, to capture the weak effect of the cylindrical gaps. The straight lines show the expected Gaussian behavior $(q∕2)$ and behavior for large $q$ with planar $(q-1)$ and cylindrical $(q-2)$ gaps.