Discrete models of force chain networks

A fundamental property of any material is its response to a localized stress applied at a boundary. For granular materials consisting of hard, cohesionless particles, not even the general form of the stress response is known. Directed force chain networks (DFCNs) provide a theoretical framework for addressing this issue, and analysis of simplified DFCN models reveal both rich mathematical structure and surprising properties. We review some basic elements of DFCN models and present a class of homogeneous solutions for cases in which force chains are restricted to lie on a discrete set of directions.Comment: 17 pages, 6 figures, dcds-B.cls; Minor corrections to version 2, but including an important factor of 2; Submitted to Discrete and Continuous Dynamical Systems B for special issue honoring David Schaeffe

Controlling spatiotemporal dynamics with time-delay feedback

We suggest a spatially local feedback mechanism for stabilizing periodic orbits in spatially extended systems. Our method, which is based on a comparison between present and past states of the system, does not require the external generation of an ideal reference state and can suppress both absolute and convective instabilities. As an example, we analyze the complex Ginzburg-Landau equation in one dimension, showing how the time-delay feedback enlarges the stability domain for travelling waves.Comment: 4 pages REVTeX + postscript file with 3 figure

Forcing nonperiodicity with a single tile

An aperiodic prototile is a shape for which infinitely many copies can be arranged to fill Euclidean space completely with no overlaps, but not in a periodic pattern. Tiling theorists refer to such a prototile as an "einstein" (a German pun on "one stone"). The possible existence of an einstein has been pondered ever since Berger's discovery of large set of prototiles that in combination can tile the plane only in a nonperiodic way. In this article we review and clarify some features of a prototile we recently introduced that is an einstein according to a reasonable definition. [This abstract does not appear in the published article.]Comment: 18 pages, 10 figures. This article has been substantially revised and accepted for publication in the Mathematical Intelligencer and is scheduled to appear in Vol 33. Citations to and quotations from this work should reference that publication. If you cite this work, please check that the published form contains precisely the material to which you intend to refe

Quasicrystalline structure of the Smith monotile tilings

We show that the tilings of the plane with the Smith hat aperiodic monotile (and its mirror image) are quasicrystals with hexagonal (C6) rotational symmetry. Although this symmetry is compatible with periodicity, the tilings are quasiperiodic with an incommensurate ratio characterizing the quasiperiodicity that stays locked to the golden mean as the tile parameters are continuously varied. Smith et al. [arXiv:2303.10798 (2023)] have shown that the hat tiling can be constructed as a decoration of a substitution tiling employing a set of four "metatiles." We analyze a modification of the metatiles that yields a set of "Key tiles," constructing a continuous family of Key tiles that contains the family corresponding to the Smith metatiles. The Key tilings can be constructed as projections of a subset of 6-dimensional hypercubic lattice points onto the two-dimensional tiling plane, and when projected onto a certain 4-dimensional subspace this subset uniformly fills four equilateral triangles. We use this feature to analytically compute the diffraction pattern of a set of unit masses placed at the tiling vertices, thereby establishing the quasiperiodic nature of the tiling. We comment on several unusual features of the family of Key tilings and hat decorations and show the tile rearrangements associated with two tilings that differ by an infinitesimal phason shift.Comment: 12 pages, 16 figure

Local growth of icosahedral quasicrystalline tilings

Icosahedral quasicrystals (IQCs) with extremely high degrees of translational order have been produced in the laboratory and found in naturally occurring minerals, yet questions remain about how IQCs form. In particular, the fundamental question of how locally determined additions to a growing cluster can lead to the intricate long-range correlations in IQCs remains open. In answer to this question, we have developed an algorithm that is capable of producing a perfectly ordered IQC, yet relies exclusively on local rules for sequential, face-to-face addition of tiles to a cluster. When the algorithm is seeded with a special type of cluster containing a defect, we find that growth is forced to infinity with high probability and that the resultant IQC has a vanishing density of defects. The geometric features underlying this algorithm can inform analyses of experimental systems and numerical models that generate highly ordered quasicrystals.Comment: 13 pages, 15 figures, 1 tabl

Hierarchical freezing in a lattice model

A certain two-dimensional lattice model with nearest and next-nearest neighbor interactions is known to have a limit-periodic ground state. We show that during a slow quench from the high temperature, disordered phase, the ground state emerges through an infinite sequence of phase transitions. We define appropriate order parameters and show that the transitions are related by renormalizations of the temperature scale. As the temperature is decreased, sublattices with increasingly large lattice constants become ordered. A rapid quench results in glass-like state due to kinetic barriers created by simultaneous freezing on sublattices with different lattice constants.Comment: 6 pages; 5 figures (minor changes, reformatted