scholarly journals Extreme boundary conditions and random tilings

2021 ◽  
Author(s):  
Jean-Marie Stéphan
Keyword(s):  
Boundary Conditions ◽  
Quantum Particle ◽  
Condensed Matter ◽  
Two Dimensions ◽  
Basic Knowledge ◽  
Dimer Model ◽  
Long Range Correlations ◽  
Statistical Mechanical ◽  
Local Densities ◽  
Random Tilings

Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.]

THE EXACT SOLUTION OF SOME LATTICE STATISTICS MODELS WITH FOUR STATES PER SITE

1964 ◽  
Vol 42 (8) ◽  
pp. 1564-1572 ◽  
Author(s):  
D. D. Betts
Keyword(s):  
Partition Function ◽  
Nearest Neighbor ◽  
Quaternary Alloy ◽  
Interesting Property ◽  
Two Dimensions ◽  
Lattice Statistics ◽  
Critical Temperatures ◽  
Different Types ◽  
Interacting Systems ◽  
Statistical Mechanical

Statistical mechanical ensembles of interacting systems localized at the sites of a regular lattice and each having four possible states are considered. A set of lattice functions is introduced which permits a considerable simplification of the partition function for general nearest-neighbor interactions. The particular case of the Potts four-state ferromagnet model is solved exactly in two dimensions. The order–disorder problem for a certain quaternary alloy model is also solved exactly on a square net. The quaternary alloy model has the interesting property that it has two critical temperatures and exhibits two different types of long-range order. The partition function for the spin-3/2 Ising model on a square net is expressed in terms of graphs without odd vertices, but has not been solved exactly.

scholarly journals Boundary states for chiral symmetries in two dimensions

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Philip Boyle Smith ◽  
David Tong
Keyword(s):  
Boundary Conditions ◽  
Ground State ◽  
Zero Mode ◽  
Central Charge ◽  
Two Dimensions ◽  
Dirac Fermions ◽  
Chiral Symmetries ◽  
Boundary States ◽  
Ground State Degeneracy ◽  
Left And Right

Abstract We study boundary states for Dirac fermions in d = 1 + 1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (−1)F , which are characterised by the existence of an unpaired Majorana zero mode.

Boundary conditions, entropy and the signature of random tilings

1996 ◽  
Vol 29 (21) ◽  
pp. 6709-6716 ◽  
Author(s):  
Dieter Joseph ◽  
Michael Baake
Keyword(s):  
Boundary Conditions ◽  
Random Tilings

Dry Aligning Dilute Active Matter

2020 ◽  
Vol 11 (1) ◽  
pp. 189-212 ◽  
Author(s):  
Hugues Chaté
Keyword(s):  
Long Range ◽  
Collective Motion ◽  
Local Level ◽  
Orientational Order ◽  
Two Dimensions ◽  
Active Matter ◽  
Long Range Correlations ◽  
Active Particles ◽  
Growing Domain

Active matter physics is about systems in which energy is dissipated at some local level to produce work. This is a generic situation, particularly in the living world but not only. What is at stake is the understanding of the fascinating, sometimes counterintuitive, emerging phenomena observed, from collective motion in animal groups to in vitro dynamical self-organization of motor proteins and biofilaments. Dry aligning dilute active matter (DADAM) is a corner of the multidimensional, fast-growing domain of active matter that has both historical and theoretical importance for the entire field. This restrictive setting only involves self-propulsion/activity, alignment, and noise, yet unexpected collective properties can emerge from it. This review provides a personal but synthetic and coherent overview of DADAM, focusing on the collective-level phenomenology of simple active particle models representing basic classes of systems and on the solutions of the continuous hydrodynamic theories that can be derived from them. The obvious fact that orientational order is advected by the aligning active particles at play is shown to be at the root of the most striking properties of DADAM systems: ( a) direct transitions to orientational order are not observed; ( b) instead generic phase separation occurs with a coexistence phase involving inhomogeneous nonlinear structures; ( c) orientational order, which can be long range even in two dimensions, is accompanied by long-range correlations and anomalous fluctuations; ( d) defects are not point-like, topologically bound objects.

A phase-field-based lattice Boltzmann method for moving contact line problems on curved stationary boundaries in two dimensions

2019 ◽  
Vol 30 (06) ◽  
pp. 1950044
Author(s):  
Weifeng Zhao
Keyword(s):  
Boundary Conditions ◽  
Lattice Boltzmann Method ◽  
Contact Line ◽  
Phase Field ◽  
Lattice Boltzmann ◽  
Two Dimensions ◽  
Curved Boundaries ◽  
Moving Contact Line ◽  
Moving Contact ◽  
Boltzmann Method

In this work, we propose a phase-field-based lattice Boltzmann method to simulate moving contact line (MCL) problems on curved boundaries. The key point of this method is to implement the boundary conditions on curved solid boundaries. Specifically, we use our recently proposed single-node scheme for the no-slip boundary condition and a new scheme is constructed to deal with the wetting boundary conditions (WBCs). In particular, three kinds of WBCs are implemented: two wetting conditions derived from the wall free energy and a characteristic MCL model based on geometry considerations. The method is validated with several MCL problems and numerical results show that the proposed method has utility for all the three WBCs on both straight and curved boundaries.

scholarly journals Regulating surface wrinkles using light

2020 ◽  
Vol 7 (7) ◽  
pp. 1247-1257
Author(s):  
Liangwei Zhou ◽  
Kaiming Hu ◽  
Wenming Zhang ◽  
Guang Meng ◽  
Jie Yin ◽  
...  
Keyword(s):  
Theoretical Model ◽  
Boundary Conditions ◽  
Functional Materials ◽  
Two Dimensions ◽  
Stress Distributions ◽  
Fabrication Technology ◽  
On Demand ◽  
Optical Grating ◽  
Wrinkle Structures ◽  
Basic Characteristics

Abstract Regulating existing micro and nano wrinkle structures into desired configurations is urgently necessary yet remains challenging, especially modulating wrinkle direction and location on demand. In this work, we propose a novel light-controlled strategy for surface wrinkles, which can dynamically and precisely regulate all basic characteristics of wrinkles, including wavelength, amplitude, direction and location (λ, A, θ and Lc), and arbitrarily tune wrinkle topographies in two dimensions (2D). By considering the bidirectional Poisson's effect and soft boundary conditions, a modified theoretical model depicting the relation between stress distributions and the basic characteristics was developed to reveal the mechanical mechanism of the regulation strategy. Furthermore, the resulting 2D ordered wrinkles can be used as a dynamic optical grating and a smart template to reversibly regulate the morphology of various functional materials. This study will pave the way for wrinkle regulation and guide fabrication technology for functional wrinkled surfaces.

scholarly journals Structural, Optical and Dynamic Properties of Thin Smectic Films

Crystals ◽  
2020 ◽  
Vol 10 (4) ◽  
pp. 321 ◽  
Author(s):  
Izabela Śliwa ◽  
A. V. Zakharov
Keyword(s):  
Optical Properties ◽  
Liquid Crystal ◽  
Transition Temperature ◽  
Solid Surface ◽  
Dynamic Properties ◽  
Condensed Matter ◽  
Theoretical Description ◽  
Structural And Optical Properties ◽  
Statistical Mechanical ◽  
Crystal Phases

The problem of predicting structural and dynamic behavior associated with thin smectic films, both deposited on a solid surface or stretched over an opening, when the temperature is slowly increased above the bulk transition temperature towards either the nematic or isotropic phases, remains an interesting one in the physics of condensed matter. A useful route in studies of structural and optical properties of thin smectic films is provided by a combination of statistical–mechanical theories, hydrodynamics of liquid crystal phases, and optical and calorimetric techniques. We believe that this review shows some useful routes not only for the further examining of the validity of a theoretical description of thin smectic films, both deposited on a solid surface or stretched over an opening, but also for analyzing their structural, optical, and dynamic properties.

Boundary conditions on a finite grid: Applications with pseudospectral wave propagation

Geophysics ◽  
1997 ◽  
Vol 62 (5) ◽  
pp. 1544-1557 ◽  
Author(s):  
Huatao Wu ◽  
Jonathan M. Lees
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Boundary Conditions ◽  
Optimal Parameter ◽  
Two Dimensions ◽  
New Method ◽  
Head Waves ◽  
Vertical Fault ◽  
Reduction Function ◽  
Grid Nodes

A new method for calculating boundary conditions at the free surface and along absorbing boundaries of a finite grid is presented. A finite, twice differentiable reduction function that achieves a 99% reduction over three wavelengths is proposed and tested. In the context of pseudospectral wave propagation, this implies a boundary layer of at least six grid nodes. The method is analyzed in one and two dimensions and the problems of waves impinging on corners are addressed. The reduction function recommended is [Formula: see text] where α is a parameter to be determined by optimization. Tests of the performance of the new method versus other common schemes are presented and analyzed. We provide a strategy for determining the optimal parameter in the reduction function. Synthetic Rayleigh waves are observed at the free surface of the simulation. Experiments with a vertical fault plane show the presence of direct, reflected, transmitted, and head waves. The presence of head waves may be used to analyze velocity contrasts across fault zones.

Sign in / Sign up

Export Citation Format

Share Document