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

Abstract

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 big", 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.PACS numbers: 05.60. Cd, 05.45.Jn, 05.45.Pq, 66.10.cg The Lorentz gas is a classical model of transport processes, in which a cloud of non-interacting point particles (modelling electrons) undergo free motion between elastic collisions with fixed hard spheres (atoms) [1]. It has been much studied as a model system for which the programme of statistical physics can be carried out in detail: to relate the known microscopic dynamics to the macroscopic behavior of the system, which in this case is diffusive [2][3][4].When the scatterers are arranged in a periodic crystal structure, the dynamics of this billiard model can be reduced to a single unit cell [2]. The curved shape of the scatterers implies that nearby trajectories separate exponentially fast, so that the system is hyperbolic (chaotic) and ergodic [5].In two dimensions, it has been shown that the cloud of particles in the periodic Lorentz gas undergoes normal diffusion, provided that the geometrical finite horizon condition is satisfied: particles cannot travel arbitrarily far without colliding with a scatterer [5,6]. By normal diffusion, we mean that the distribution of particle positions behaves like solutions of the diffusion equation; in particular, the mean-squared displacement (variance) grows asymptotically linearly in time:r(t) 2 ∼ 2dDt when t → ∞, where r(t) is the displacement of a particle at time t from its initial position, d is the spatial dimension, · denotes a mean over initial conditions, and the diffusion coefficient D gives the asymptotic growth rate. When the horizon is infinite, however, particles can undergo arbitrarily long free flights along corridors in the structure. It was long argued [7][8][9] and has recently been proved [10], that there is then weak superdiffusive behavior, with r(t) 2 ∼ t logt, so that the diffusion coefficient no longer exists.For higher-dimensional periodic Lorentz gases, rigorous results on ergodic properties [11] and diffusive properties [12] have been obtained; recent progress in their analysis has been made [13,14], including in the limit of small scatterers [15]. In particular, higher-dimensional Lorentz gases are believed to exhibit normal diffusion when the horizon is finite [12].Nonetheless, the study of billiard models in higher dimensions, especially three dimensions, has received surprisingly little attention from the physics community, despite their interest as simple ...

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

“…(3). We note that there has been recent interest in the extension of these results to higher-dimensional Lorentz gases [19][20][21], where additional effects come into play.…”

“…We refer * giampaolo.cristadoro@unibo.it † thomas.gilbert@ulb.ac.be ‡ marco.lenci@unibo.it § dpsanders@ciencias.unam.mx to these regions as corridors, following Ref. [18]; they are elsewhere termed gaps [19], horizons [20], and free planes [21].…”

Emergence of distinguishability of patterns of collisions of particles in a non-equilibrium chaotic system

“…Ref. [15], this is not a limitation for the analysis below, that concerns the most probable events. The Langevin equations (3) produce non-trivial physical conclusions [2].…”

Embedding Quasicrystals in a Periodic Cell: Dynamics in Quasiperiodic Structures

“…1 for the meansquared displacement [19][20][21][22][23]; here, ÁxðtÞ : ¼ xðtÞ À xð0Þ is the displacement of a particle at time t, and hÁi denotes an average over uniform initial conditions. However, a key role is played by the presence of channels, i.e., empty regions through which particles may travel infinitely far without colliding: channels of the highest dimension, n À 1, give rise to weak superdiffusion of the form hÁxðtÞ 2 i $ t lnt, with a logarithmic correction [23][24][25][26][27].…”

“…Suppose that the radius r of the obstacle B, and hence of the cylinder K, is small enough that there is (at least) one face of the cube which is not intersected by any of the three cylindrical pieces of K. Then in the 3D billiard with arbitrary velocities (not restricted to be parallel to E) there is a planar channel Å [23] lying along any such face. Now restricting particle velocities to be parallel to E, we see that each planar channel Å in the 3D periodic billiard induces a rectangular channel in the 2D quasiperiodic LG, given by the intersection of E with Å.…”