Horizons and free path distributions in quasiperiodic Lorentz gases
arXiv:1511.00340 · doi:10.1103/PhysRevE.92.052131
Abstract
We study the structure of quasiperiodic Lorentz gases, i.e., particles bouncing elastically off fixed obstacles arranged in quasiperiodic lattices. By employing a construction to embed such structures into a higher dimensional periodic hyperlattice, we give a simple and efficient algorithm for numerical simulation of the dynamics of these systems. This same construction shows that quasiperiodic Lorentz gases generically exhibit a regime with infinite horizon, that is, empty channels through which the particles move without colliding, when the obstacles are small enough; in this case, the distribution of free paths is asymptotically a power law with exponent -3, as expected from infinite-horizon periodic Lorentz gases. For the critical radius at which these channels disappear, however, a new regime with locally-finite horizon arises, where this distribution has an unexpected exponent of -5, previously observed only in a Lorentz gas formed by superposing three incommensurable periodic lattices in the Boltzmann-Grad limit where the radius of the obstacles tends to zero.
arXiv comment9 pages, 7 figures
References in corpus (21)
- Disordered, Quasicrystalline and Crystalline Phases of Densely Packed Tetrahedra
- Steady-State Electrical Conduction in the Periodic Lorentz Gas
- Optical extinction due to intrinsic structural variations of photonic crystals
- Localization Transition of the Three-Dimensional Lorentz Model and Continuum Percolation
- Controlled self-assembly of periodic and aperiodic cluster crystals
- Crossover in the Slow Decay of Dynamic Correlations in the Lorentz Model
- New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis
- Free path lengths in quasicrystals
- Chiral Quasicrystalline Order and Dodecahedral Geometry in Exceptional Families of Viruses
- Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards
- Power-law distributions for the free path length in Lorentz gases
- Fine structure of distributions and central limit theorem in diffusive billiards
- Free path lengths in quasi crystals
- Typical recurrence for the ehrenfest wind-tree model
- Diffusive properties of persistent walks on cubic lattices with application to periodic Lorentz gases
- Tail asymptotics of free path lengths for the periodic Lorentz process. On Dettmann's geometric conjectures
- Normal diffusion in crystal structures and higher-dimensional billiard models with gaps
- Random walk approach to the d-dimensional disordered Lorentz gas
- Systematic Density Expansion of the Lyapunov Exponents for a Two-dimensional Random Lorentz Gas
- 12-fold Quasicrystallography from affine F4, B6, and E6
- Invariance principle for the periodic Lorentz gas in the Boltzmann-Grad limit